Structural Characteristics of Two-Sender Index Coding

This paper studies index coding with two senders. In this setup, source messages are distributed among the senders possibly with common messages. In addition, there are multiple receivers, with each receiver having some messages a priori, known as side-information, and requesting one unique message such that each message is requested by only one receiver. Index coding in this setup is called two-sender unicast index coding (TSUIC). The main goal is to find the shortest aggregate normalized codelength, which is expressed as the optimal broadcast rate. In this work, firstly, for a given TSUIC problem, we form three independent sub-problems each consisting of the only subset of the messages, based on whether the messages are available only in one of the senders or in both senders. Then, we express the optimal broadcast rate of the TSUIC problem as a function of the optimal broadcast rates of those independent sub-problems. In this way, we discover the structural characteristics of TSUIC. For the proofs of our results, we utilize confusion graphs and coding techniques used in single-sender index coding. To adapt the confusion graph technique in TSUIC, we introduce a new graph-coloring approach that is different from the normal graph coloring, which we call two-sender graph coloring, and propose a way of grouping the vertices to analyze the number of colors used. We further determine a class of TSUIC instances where a certain type of side-information can be removed without affecting their optimal broadcast rates. Finally, we generalize the results of a class of TSUIC problems to multiple senders.

broadcast rate formulation of the problems, the senders are connected to receivers via noiseless broadcast links (orthogonal to each other) with flexible capacities, and we aim to characterize the total transmitted bits per received message bits of those problems. On the other hand, in the capacity region formulation of the problems, the senders are connected to receivers via noiseless broadcast links of arbitrary fixed capacities, and we aim to characterize the closure of the set of all achievable rate tuples 1 of messages of those problems.
The multi-sender index-coding problem was first studied by Ong et al. [15]. They considered the broadcast rate formulation of the problems, and presented results by implementing a graphtheoretic approach (which requires finite length messages). At the senders' end, they considered multiple senders each knowing only a subset of the messages. At the receivers' end, they considered a single-uniprior message setting, meaning each receiver knows only one message requested by some other receiver a priori, one message is known to only one receiver, and one receiver may request multiple messages. For this setup, they devised lower and upper bounds on the optimal broadcast rate. 2 The results were established using information-flow graphs, which represent receivers' request, and message graphs, which represent senders' message setting.
Furthermore, they showed that their upper and lower bounds coincide for a class of problem instances where no two senders have messages in common. In another work, Thapa et al. [16], considered a similar model (to the model considered by Ong et al.) but with a unicast message setting, meaning each message is requested by only one receiver, each receiver requests only one message, and each receiver knows a subset of messages requested by other receivers a priori.
Based on a graph-theoretic approach, they established upper bounds on the optimal broadcast rate. In particular, they focused on the two-sender case, called two-sender unicast index coding (TSUIC). They extended existing single-sender index-coding schemes, namely the cycle-cover scheme [17], [18], the clique-cover scheme [1], [2] and the local-chromatic scheme [10] to the corresponding schemes in TSUIC.
So far, works on the capacity region formulation of the problems use a random-coding approach (which requires infinitely long messages). Inner and outer bounds on the capacity region have 1 Consider a message xi of ti bits requested by receiver i. A rate tuple of messages is achievable if there exists an index code such that each rate Ri in the tuple is less than or equal to the respective message bits ti per total transmitted bits. 2 The total transmitted bits per received message bits is referred to as the broadcast rate. The optimal broadcast rate is the minimum over all possible broadcast rates for an index coding instance. It is also the reciprocal of the symmetric capacity, the supremum of equal achievable rates. finite fields) are less explored. For an index coding instance in the unicast message setting and the non-asymptotic regime in the message size, our techniques in this paper can be used to upper bound the optimal broadcast rate of this instance by a function of the optimal broadcast rates of its sub-instances in single-sender unicast index coding.
This study helps us answer some basic questions as follows: • Given a TSUIC instance, what is its optimal rate of transmission?
• Under what conditions distributed encoding in TSUIC achieves the same optimal broadcast rate of centralized encoding?
• Which side-information are not useful in the sense that we can remove it without changing the optimal broadcast rate of TSUIC problems?
In particular, this paper contributes the following for TSUIC: 1) Introducing two-sender graph coloring of confusion graphs in TSUIC (Definition 8, Section V): In TSUIC problems, as the two senders (encoders) contain some messages in common, the standard way of graph coloring of the confusion graph may not lead us to an index code. In this regard, we need a different kind of coloring function in TSUIC, and thus, in this paper, we propose a novel way of coloring dedicated especially to color the confusion graphs in TSUIC.
2) Presenting a way of grouping the vertices of confusion graphs in TSUIC (Section VI): By exploiting the symmetry of the confusion graph, we propose a way of grouping its vertices for analysis purposes mainly in its two-sender graph coloring. In particular, this grouping helps us to analyze the number of colors used in two-sender graph coloring of a confusion graph.
3) Deriving the optimal broadcast rates of TSUIC problems as a function of the optimal broadcast rates of its sub-problems (Theorem 4-9): We divide a TSUIC problem into three independent sub-problems based on the requested messages by receivers, whether the messages are present in only one of the senders or in both senders. Now in TSUIC, considering the interactions between these three independent sub-problems, we derive the optimal broadcast rate (in both asymptotic and non-asymptotic regimes in the message size) of the problem as a function of the optimal broadcast rates of its sub-problems. Moreover, we bound the optimal broadcast rate, and show that the bounds are tight for classes of TSUIC instances (sometimes with conditions). Besides, we find a class of TSUIC instances 1. An example of a TSUIC problem with four receivers: The total message set M = M1 ∪ M2 is distributed among two senders S1 and S2. Each sender is connected to all receivers via a noiseless broadcast channel. Each receiver, r ∈ {1, 2, 3, 4}, having some side-information, which is represented by a set Hr, requests a unique message xr. We assume that each sender Ss, s ∈ {1, 2}, is broadcasting a sub-codeword Cs of length |Cs|, and they cooperate with each other to reduce their aggregate transmissions. Precisely, we aim to find a two-sender index code with the minimum sum of lengths |C1| + |C2|.
This means each sender S s encodes its known messages to a p s -bit sub-codeword, for some non-negative integer p s . We assume that each receiver r receives sub-codewords from both of the two senders without any noise, and decodes x r from the received sub-codewords and H r . Now we define the minimum encoding length, which measures the performance of a code (C 1 , C 2 ), in the following. The optimal broadcast rate over all t is defined as β inf t β t = lim t→∞ β t . The limit exists and equal to the infimum due to the subadditivity of tβ t = p 1 + p 2 and Fekete's lemma [24].
Remark 1: With the (optimal) broadcast rate as a performance metric, we can treat SSUIC as a special case of TSUIC when M 1 = M or M 2 = M. Furthermore, for this case, the sender with M alone will be responsible to fulfill the demands made by all receivers.

B. Representation of the receivers' side-information and the senders' message setting of TSUIC problems
An index-coding problem can be modeled by graphs, which are defined as follows: Definition 4 (Directed graphs and undirected graphs): A directed graph is an ordered pair From now on in this paper, we call directed graphs simply digraphs, and undirected graphs simply graphs.
The receivers' message setting of a UIC problem is represented by a side-information digraph is the out-neighborhood of a vertex i in D. In this paper, for convenience, a receiver i is also referred to as a vertex i, and vice versa. We also use the compact form of representation of an instance of UIC problems as used by Arbabjolfaei et al. [7], where a sequence (i|N + D (i)), for all i ∈ V (D), represents a UIC problem. In TSUIC, S 1 (sender one) encodes the messages in M 1 , and S 2 (sender two) encodes the messages in M 2 in order to achieve the optimal broadcast rate β t . In general, each sender has private messages and common messages defined as follows: Let P 1 M 1 \ M 2 and P 2 M 2 \ M 1 be the set of private messages at senders S 1 and S 2 , respectively, and P 3 M 1 ∩ M 2 be the set of common messages at both senders. Now for a given side-information digraph D, without loss of generality, we define the following vertex-induced sub-digraphs those dividing D in to three parts: For i ∈ {1, 2, 3}, let D i be the sub-digraph of D induced by vertices {j : x j ∈ P i }. We refer to D 1 , D 2 and D 3 as per this definition throughout this paper unless stated otherwise. From the definition, it is clear that D 1 , D 2 and D 3 are the three sub-digraphs i, k ∈ {1, 2, 3}. In TSUIC, the senders are limited to transmit only their messages, and this limitation is defined formally as a constraint due to the two senders, and it is defined as follows: Definition 5 (Constraint due to the two senders): The constraint due to the two senders is the following: While encoding, any two private messages x i ∈ P 1 and x j ∈ P 2 should not be encoded together (with or without other messages) to construct one coded symbol, or alternatively any two-sender index code can be written as (C 1 , C 2 ) such that C 1 = F 1 (M\P 2 ) and C 2 = F 2 (M\P 1 ).
In TSUIC, to reflect the senders' message setting, we introduce an undirected graph, denoted , that is constructed in the following way:

III. A NOVEL WAY OF INVESTIGATING TSUIC PROBLEMS
In a TSUIC problem, if there is no common message, i.e., P 3 = ∅, then in our earlier work, we have proved that the problem is equivalent to two separate SSUIC problems ([16, Theorem 1]).
However, if P 3 = ∅, then the problem is less well understood. We propose to understand this problem by dividing it into three sub-problems based on the type of messages at the senders (whether they are common or private), and then study the interactions among these sub-problems due to the side-information present at the receivers. In this way, we can devise the structural characteristics of TSUIC problems. For a given problem D, the three sub-problems based on the type of messages are D 1 , D 2 and D 3 . The side-information present at receivers of one sub-problem about messages requested by receivers of other sub-problems are formally referred to as an interaction between those sub-problems, defined in the following. We will see that this allows us to derive β t (D, there exists an arc from a vertex of D i to a vertex of D j for any i, j ∈ {1, 2, 3}, i = j, then In other words, all the vertices of the sub-digraph D i interact in the same way to all the vertices of the sub-digraph D j . For an example of a fully-participated interaction see Fig. 2a. If an interaction between the sub-digraphs is not a fully-participated interaction, then it is called a partially-participated interaction between the sub-digraphs. For example of a partially-participated interaction see Fig. 2b. For any D, if  is further classified into smaller sub-cases II-A, II-B, II-C, II-D and II-E. Refer to Fig. 3 for details, where a case of H is labeled by H i for i ∈ {1, 2, . . . , 64}. Note that an interaction between D 1 , D 2 and D 3 of D defines arcs between them (not within the sub-digraph), and the cases of interactions (acyclic or cyclic) are defined with respect to the orientation of the arcs between the sub-digraphs. In this paper, a fully-participated interaction and a partially-participated interaction between D 1 , D 2 and D 3 of D are called a cyclic-fully-participated interaction and a cyclic-partially-participated interaction between the sub-digraphs, respectively, if and only if f (D) has some cycles (for example, see CASE II in Fig. 3).
In SSUIC, Arbabjolfaei and Kim [8,Prop. 1] argued that the structural properties can reduce the number of problems that need to be studied. This paper investigates the structural characteristics of TSUIC problems for the same purpose by studying the interactions among D 1 , D 2 and D 3 of D. Moreover, structural properties can be used to determine the criticality/non-criticality of arcs in TSUIC as in its SSUIC counterpart [9], [25]. In this work, an arc is said to be critical if removing the arc strictly increases the optimal broadcast rate. This paper, in TSUIC, analyzes all cases of fully-participated and some cases of partially-participated interactions between D 1 , D 2 and D 3 of D, and establishes their optimal broadcast rates as a function of the optimal broadcast rates of D 1 , D 2 and D 3 . This is done by utilizing existing SSUIC's results and our proposed coloring of confusion graphs for TSUIC, which we discuss in the subsequent sections. First, we define some notations, which we use throughout this paper, in the following section.
3) Consider that bits are associated with the messages requested by vertices of D i , i ∈ {1, 2, 3}.
Each realization of the bits, i.e., each member in {0, 1} tn i is represented by a unique label Fig. 9a in Appendix A outlines each tuple b j i D i for t = 1. Each message tuple (x 1 , . . . , x N ) can then be uniquely written as is similarly defined. a) Let C(D i ) be an index code (linear or non-linear) having a codeword length of p(C(D i )) bits (i.e., |C(D i )| = p(C(D i ))), for a given t (message bits), that achieves ) be two parts of the sequence of bits of a codeword of C(D i ) such that 1} t for all j ∈ {1, 2, . . . , N }. We say that two tuples are confusable if they are confusable at some receiver r. Clearly, in an index coding, we cannot encode message tuples that are confusable to the same codeword; otherwise one of the receivers cannot decode its requested message successfully. The confusability among all possible N -tuples of messages (each message having t bits) for an index-coding problem is represented by a graph called confusion graph, and it is defined as follows: Consider coloring a confusion graph Γ t (D) with a set of colors J . Now we get a family of sets of independent vertices where all vertices belonging to one set are assigned with the same color in the graph coloring. Here a set of independent vertices refers to a vertex set where any pair of vertices are not connected by an edge in Γ t (D), and we call such a set an independent vertex set. The tuples representing vertices within an independent vertex set are not confusable, and hence they can be coded into the same codeword. Assigning each independent vertex set (whose vertices are all colored by a unique color) a unique codeword provides us a valid index code having |J | codewords. Thus there exists a bijective mapping I : J → C, where C is an index code (or a set of codewords that satisfies the demands made by all receivers). We know that χ (Γ t (D)) = min J |J |. In SSUIC, it is shown that the optimal broadcast rate of an index-coding problem D with t-bit messages can be obtained by using confusion graphs. This is stated in the following theorem.
The notion of confusion graphs has been considered in the index coding literature, and it has been shown to be an effective tool for proving important results, for example, Bar-Yossef et al. [3], Alon et al. [23], and Arbabjolfaei et al. [8] in their respective works, referred to the confusion graph for the proof of results related to the odd hole and the odd anti-hole [3], the gap between β and β t=1 of hypergraphs [23], and the structural properties of the index-coding problems [8], respectively.
The confusion graph, which is only a function of the side-information graph, does not depend on the number of senders. Its coloring function described above for SSUIC may not lead to an index code for TSUIC due to the source constraint. In this work, we propose a way of coloring the confusion graphs in TSUIC, we call this two-sender graph coloring. Now we present the formal definition of two-sender graph coloring of confusion graphs for TSUIC.
). Remark 2: The two-sender graph coloring is not a b-fold coloring that assigns a set of b colors to each vertex such that the color sets corresponding to two adjacent vertices do not share any color (refer to the definition of the fractional graph coloring [8]). In our definition, the color sets can share colors, as long as the color vectors (i.e., ordered pairs) are different. Now in the form of lemmas, we discuss two-sender graph coloring of Γ t (D) in detail. Before this, we first assume the following: For any indices i , j , assume that c i and c j are any two distinct colors if i = j , and let (c i , c j ) be an ordered pair of colors. Any two ordered pairs of colors, (c i 1 , c j 1 ) and (c i 2 , c j 2 ) for indices i 1 , j 1 , i 2 , j 2 , are said to be different (or not equal) if and only if i 1 = i 2 or j 1 = j 2 or both. If a color c i is associated to a sender S s , s ∈ {1, 2}, then we denote it by c s i . In TSUIC, the two senders encode separately, so in the aforementioned definition, we need to assign an ordered pair of colors for each vertex, where the first color is associated with S 1 and the second color with S 2 . Now we have the following lemmas.
Lemma 1: For any two vertices, u N ,v N ∈ V (Γ t (D)) such that they are labeled by for some indices i 1 , i 2 , j 1 , j 2 .
Proof: It is given that (u N , v N ) ∈ E(Γ t (D)), so u N and v N are confusable. Moreover, these two tuples are confusable only at some vertex in V (D 1 ). This is because the labels of u N and v N , respectively, are different only in b i D 1 sub-label (which is representing tn 1 -bit tuples of the messages requested by vertices in V (D 1 )). Now for the sender S 2 , which does not contain any message in P 1 (messages requested by receivers in V (D 1 )), provides the same color to both vertices. Thus c 2 On the other hand, for the sender S 1 , which contains all messages in P 1 , it is necessary to have In a similar reasoning as in the above proof (of Lemma 1), one can prove the following lemmas: If u N and v N are confusable at some vertices in V (D 1 ) and in V (D 2 ), then referring to Lemma 1 and 2, we get the following: If u N and v N are confusable at some vertices in V (D 3 ), then while coloring Γ t (D) in two-sender graph coloring, it suffices to have a different color associated with any one of the senders because all the messages in P 3 are contained by both senders S 1 and S 2 . Thus we have the following lemma: ) and Now for a TSUIC problem with t-bit messages, we have the following theorem: Theorem 2: The optimal broadcast rate for a TSUIC problem with t-bit messages is Proof: For s ∈ {1, 2}, consider J s , a coloring function of the sender S s , with a set of colors J s such that the ordered pairs of colors in {(c 1 i , c 2 j ) : c 1 i ∈ J 1 , and c 2 j ∈ J 2 }, i ∈ {1, 2, . . . , |J 1 |} and j ∈ {1, 2, . . . , |J 2 |}, properly color the vertices of Γ t (D) in two-sender graph coloring. A two-sender index code is obtained by S 1 mapping distinct colors in J 1 to distinct sub-codewords, and S 2 mapping distinct colors in J 2 to distinct sub-codewords. By definition, all confusable vertex pairs are assigned a different codewords. Now for s ∈ {1, 2}, the sender S s transmits |J s | sub-codewords. Equivalently, log 2 |J s | bits are transmitted by S s . This is because the number of bits required to index |J s | colors are log 2 |J s | . Minimizing the sum ( log 2 |J 1 | + log 2 |J 2 | ) over all coloring functions J 1 (of S 1 ) and J 2 (of S 2 ) per received message bits (i.e., t), we get t , so there exists a two-sender index code such that S 1 and S 2 transmit p 1 -bit and p 2 -bit sub-codewords, respectively, resulting in Now for each sender S s , we know that there are at most 2 p s possible sub-codewords. Assume a bijective function that maps each sub-codeword to a color. This function provides us 2 p s or fewer distinct colors associated with each sender S s . Moreover, S s properly colors Γ t (D) with 2 p s or fewer distinct colors in the two-sender graph coloring; otherwise, the two-sender index code, which is formed by p 1 -bit and p 2 -bit sub-codewords of S 1 and S 2 , respectively, will be an invalid code (contradicts our assumption). Altogether, there exists a valid two-sender graph coloring such that |J 1 | ≤ 2 p 1 and |J 2 | ≤ 2 p 2 , or equivalently, p 1 ≥ log 2 |J 1 | and p 2 ≥ log 2 |J 2 | as both are non-negative integers. Substituting the inequalities of p 1 and p 2 in (4), wet get Now we prove equality in (3). This is done by contradiction. Suppose that From (5) and (6), we get and this leads to a contradiction. Thus β t (D, G o ) = min We illustrate two-sender graph coloring of a confusion graph in TSUIC, and a mapping function that maps colors to codewords at each sender from the following example.
Example 1: Consider a TSUIC problem (D, G o ) of the following: (1|2), (2|1), (3|1), and For example, (0, 0, 0) and (1, 0, 0) are connected by an edge because these two message tuples are confused at receiver 1. The confusion graph Γ 1 (D) is depicted in Fig. 4b. Now we perform two-sender graph coloring of the vertices of Γ 1 (D). In two-sender graph coloring, each vertex of Γ 1 (D) is assigned with an ordered pair of colors; the first color is always associated with S 1 and the second color is always associated with S 2 , and we color the vertices as dictated by Lemma 1 to Lemma 4. For example, consider (0, 0, 0) and (1, 0, 0). These two tuples are confused at receiver 1 (requesting x 1 ). As S 2 , which does not know x 1 , the tuples (0, 0, 0) and (1, 0, 0), which have the same second and third message bits, are treated as the same. Thus S 2 must assign the same color, say RED, to both the tuples. As S 1 knows x 1 and the tuples are confusable at receiver 1, it must assign two different colors, say RED and BLUE, to (0, 0, 0) and and that must be transmitted by that sender, so there must be at least one transmission by that In this section, we have introduced confusion graphs. As this method has been utilized for only SSUIC, we propose an approach to use the confusion graph in TSUIC to study structural characteristics of TSUIC problems. As a confusion graph possesses some symmetry within -in fact, all confusion graphs are vertex-transitive 4 -while analyzing them (especially coloring), we systematically group its vertices and then analyze the graph based on these groups (rather than individual vertices). This way, for a TSUIC problem whose sub-problems interact with each other in some way, we can reduce the complexity arising during its analysis (especially finding the number of colors in a proper coloring of Γ t (D)) due to the number of vertices, which is exponential in t and N . In the following section, we introduce our proposed method of grouping the vertices of Γ t (D).

VI. GROUPING THE VERTICES OF Γ t (D): STATING ITS CHARACTERISTICS
A vertex of the confusion graph Γ t (D) is represented by a tuple x N that is labeled by a unique Fig. 5). For the ease of analysis, we collect all the vertices with the same b k D 3 sub-label into one group, and the subgraph of Γ t (D) induced by that group of vertices is called the k-th block (refer to Fig. 5 for the functional block diagram). Moreover, this grouping provides . Clearly, all blocks in Γ t (D) are isomorphic graphs. This is because each sub-labels are different only for different blocks), and the edges in any block is due to the confusion at some receivers in V (D 1 ∪ D 2 ). Moreover, the (tn 1 + tn 2 )-bit tuples of messages requested by the vertices in . Now we further group the vertices of a block with the same b k D 3 sub-label in two ways: (i) Grouping all vertices with the same b j D 2 sub-label into one sub-group, and the sub-graph of Γ t (D) induced by that group of vertices is called the j-th sub-block. This sub-grouping of vertices provides 2 tn 2 sub-blocks within each block, and its vertex set is represented by Clearly, all sub-blocks of a Γ t (D) are isomorphic graphs. This is because each sub-block consists sub-labels are different only for different sub-blocks), and the edges in any sub-block is due to the confusion at some receivers in V (D 1 ). Moreover, the tn 1 -bit tuples of messages requested by the vertices in V (D 1 ) are labeled by b i D 1 . (ii) Grouping all vertices with the same b i D 1 sub-label into one sub-group, and this sub-grouping of vertices provides 2 tn 1 sub-groups of this kind within each block, and its vertex set is represented . From a similar reasoning as presented above for the case of the sub-block, it is not difficult to see that all the vertex-induced graphs of Γ t (D), each induced by the vertices in , for all i and k, are isomorphic graphs. Now we illustrate the aforesaid notations and grouping of the vertices by an example.   (see Fig. 4b).
In addition, we have four different sub-blocks, each formed by the vertices in one of the following sets: (for a general outline refer to For a confusion graph Γ t (D), we have the following propositions after grouping its vertices according to our proposed method. These are helpful for understanding the construction of a confusion graph, and are useful for the proofs of our results in the subsequent sections.
for any be G. We know that any vertex in the set sub-label are included in the set. Thus in G, any edge between its vertices is only due to the confusion at some receiver belonging to V (D 1 ) (corresponding to the change in bits of b i D 1 sub-label of the vertices). We know that and any edge between its vertices is due to the confusion at some receiver belonging to ) ∈ E(G) and vice-versa. This is because the edges are due to the confusion of the tuples, representing those vertices, at some receiver belonging to V (D 1 ). Consequently, Γ t (D 1 ) and G are isomorphic graphs. Now we illustrate Proposition 1 by an example. Consider the TSUIC problem stated in Example 1. The confusion graph of D 1 , Γ 1 (D 1 ), has two vertices 0 and 1 connected by an edge as they are confused at receiver 1. Now the sub-graph of Γ 1 (D) induced by the vertices in connected by an edge as they are confused at receiver 1, and clearly, this vertex-induced sub-graph is isomorphic is the vertex set of the j-th sub-block of the k-th block of Γ t (D). In a similar way to the proof of Proposition 1, one can prove the following proposition.

Proposition 2:
The following are isomorphic graphs: (i) Γ t (D 2 ) and the sub-graph of Γ t (D) and any k-th block of a Γ t (D).

Proposition 3:
In two-sender graph coloring of the sub-graph of Γ t (D) induced by the vertices is the minimum number of total ordered pairs of colors required to color the vertex-induced sub-graph, and the minimum number of colors associated with S 1 and S 2 are χ (Γ t (D 1 )) and one, respectively.
sub-labels (due to our proposed method of grouping the vertices of Γ t (D)). So, the edges between the vertices of G are only due to the confusion at some receivers belonging to V (D 1 ). From Lemma 1, any pair of vertices of G connected by an edge must have different colors associated with S 1 and the same color associated with S 2 . Thus the minimum number of colors associated with S 2 is one. From Proposition 1, G is isomorphic to Γ t (D 1 ), so it has the minimum of χ (Γ t (D 1 )) independent vertex sets. Thus we need at least χ (Γ t (D 1 )) different ordered pairs of colors to color G. Moreover, in two-sender graph coloring, a proper two-sender graph coloring is to let S 1 color the vertices in the same way as we color Γ t (D 1 ). As the colors associated with S 2 is one, the minimum number of colors associated with S 1 must be χ (Γ t (D 1 )) in order to have the minimum of χ (Γ t (D 1 )) ordered pairs of colors in two-sender graph coloring.
In a similar way to the proof of Proposition 3, one can prove the following proposition.
Proposition 4: In two-sender graph coloring of the sub-graph of Γ t (D) induced by the vertices is the minimum number of total ordered pairs of colors required to color the vertex-induced sub-graph, and the minimum number of colors associated with S 1 and S 2 are one and χ (Γ t (D 2 )), respectively.
Proposition 5: In two-sender graph coloring of the sub-graph of Γ t (D) induced by the vertices , the minimum number of total ordered pairs of colors required to color the vertex-induced sub-graph is χ (Γ t (D 3 )).
Proof: We know that Γ t (D 3 ) requires the minimum of χ (Γ t (D 3 )) different colors in its coloring in SSUIC. Thus it has the minimum of χ (Γ t (D 3 )) independent vertex sets. From Proposition 2, the sub-graph of Γ t (D) induced by the vertices in , so it has the minimum of χ (Γ t (D 3 )) independent vertex sets. In two-sender graph coloring, we assign each independent vertex set a unique ordered pair of colors. Thus the vertex-induced sub-graph requires the minimum of χ (Γ t (D 1 )) ordered pairs of colors.
For convenience, we define the following types of edges of Γ t (D): is also an index code for D, but the converse is not always true, so we have the following: In TSUIC, each sender S s transmits at least β(D s ), for s ∈ {1, 2}. We now provide a lower bound of the optimal broadcast rate for a TSUIC problem with t-bit messages in the following lemma.
Lemma 6 (A lower bound): For any two-sender index-coding problem (D, Proof: For any two-sender index-coding problem (D, Th. 1]). For any index-coding problem, its broadcast rate is always lower bounded by the broadcast rate of any sub-problem, so we get We know that lim Definition 3). Now taking a limit t → ∞ on both sides in (8), we get To compute the simple lower bound to the optimal broadcast rate of a given problem in TSUIC, In the rest of this paper, for any D having fully-participated interactions between its sub-digraphs  Table I. Furthermore, similar results are presented for D whose f (D) is of CASE I and Case II-A, and it has partially-participated interactions between the sub-digraphs.

VIII. MAIN RESULTS
A. Optimal broadcast rate of a digraph D whose f (D) belongs to CASE I and CASE II-A: The arcs between D 1 , D 2 and D 3 are not critical in the asymptotic regime in the message size. {x4 ⊕ x5, x1, x2, x3}, that is obtained by transmitting x4 ⊕ x5 from S1, and x1, x2, x3 from S2, achieves the optimal broadcast rate both in TSUIC and SSUIC.
acyclic. Thus one can arrange D 1 , D 2 and D 3 in a sequence such that there is no arc between D 1 , D 2 and D 3 in a backward direction. Without loss of generality, let the sequence be D 1 , D 2 and D 3 . Now for D, referring to Theorem 3, we get From Lemma 5, we have Now from (10) and (11), we get In TSUIC, if we consider the sub-digraphs D 1 , D 2 and D 3 separately, then their respective source constraint graphs are the sub-graphs of G 0 induced by vertices V (D 1 ), V (D 2 ) and V (D 3 ), denoted G 1 0 , G 2 0 and G 3 0 , respectively. These sub-graphs are edgeless graphs, and thus one can get β(D 1 , G 1 0 ) = β(D 1 ), β(D 2 , G 2 0 ) = β(D 2 ) and β(D 3 , G 3 0 ) = β(D 3 ). We know that the optimal broadcast rate of a side-information digraph is always less than or equal to the sum of the optimal broadcast rates of its sub-digraphs, so From (10), (12) and (13) for some ∈ {−2, −1, 0}. Observe that the interaction between D 1 , D 2 and D 3 of D such that f (D) ∈ {H 2 , H 3 . . . , H 12 } has fewer interactions (between D 1 , D 2 and D 3 ) than that of D 27 , so and more interactions than that of D 1 , so Now from (14), (15) and (16) For any D whose f (D) belongs to a digraph of CASE II in Fig. 3, referring to the definition of the mapping function f , we observe that the interaction between D 1 , D 2 and D 3 of D are not acyclic. The CASE II is further divided into five sub-cases. Case II-A depicts a sub-case when there is no 3 → 1 and 3 → 2 in H, and for the digraph whose f (D) is of this sub-case, we have the following theorem.
Proof: Refer to Appendix C.
Proof: Refer to Appendix D.

IX. DISCUSSIONS
Consider any digraph D and its sub-digraphs D 1 , D 2 and D 3 . Let d + D (u) be the out-degree of a vertex u of D. Now we make the following two observations in TSUIC: 9. In the same regard, the result is not known for D having a cyclic-partially-participated interaction between its sub-digraphs D 1 , D 2 and D 3 .
• Non-critical arcs in SSUIC are not necessarily non-critical in TSUIC: We illustrate this with an example. Consider the TSUIC problem stated in Example 1 (whose f (D) = H 58 ).
In SSUIC, we know that the optimal broadcast rate β(D) = 2. This problem has an arc (3, 1) that is non-critical in SSUIC (its removal does not change the optimal broadcast rate) but it is critical in TSUIC. This can be understood from the following: In SSUIC, we can remove the arc (3, 1) ∈ A(D), and still form a valid index code {x 1 ⊕ x 2 , x 3 } that achieves β(D). This infers that removing the arc (3, 1) does not affect the optimal broadcast rate in SSUIC. However, in TSUIC, if we remove the arc (3, 1) ∈ A(D), then the new problem, say D , has β(D , G o ) = 3 (applying Theorem 6), whereas we get a valid two-sender index

X. CONCLUDING REMARKS AND OPEN PROBLEMS
In this paper, we studied two-sender unicast-index-coding problems and established their structural characteristics. Noting that SSUIC is a well studied problem (though for any arbitrary instance, it is still an open problem), there have been many important contributions made in the literature. In this paper we solved TSUIC instances by expressing the optimal broadcast rates in terms of that of SSUIC. To this end, we introduced a two-sender graph coloring of confusion graphs in TSUIC, and propose a way of grouping the vertices of a confusion graph for the ease of its analysis while coloring. With these, we derived optimal broadcast rates of TSUIC problems, both in the asymptotic and non-asymptotic regime, as a function of the optimal broadcast rates of their sub-problems. We have also presented a class of TSUIC instances where the interactions between the sub-problems of the problem are not critical. Some open problems for future works are the following: • Study of the critical edges in the TSUIC problems: It is observed that the non-critical arcs in SSUIC can be critical arcs in TSUIC. This requires further study.
• Extend the results to more than two senders: As our study of TSUIC problems is a step towards understanding multi-sender index coding, a general distributed index coding, it is left as a future work to extend the approaches implemented and the results obtained in this paper to more general setups.
• Finding the optimal broadcast rates of TSUIC problems with cyclic-partially-participated interactions: The analysis of D with partially-participated interactions between its subdigraphs D 1 , D 2 and D 3 is left as a future work.

APPENDIX A
The two figures in Fig. 9 outline the labels used to represent vertices of a confusion graph.  representation of the vertices (e.g., ) and sets of vertices (e.g., Before proving proposition 6, with the help of the following example, we provide an overview of the construction of the confusion graph and its two-sender graph coloring, which after generalization leads to the proof of Proposition 6. Here, , and its two-sender graph coloring (where each vertex is assigned with an ordered pair of colors such that the first color is always associated with S1 and the second color is always associated with S2). The edges are shown in color only for an illustration purpose, it is not an edge coloring.
We color similarly for any individual sub-block. For the sub-graph of Γ 1 (D) induced by the = {(0, 0, 00), (0, 1, 00)}, as these two tuples ((0, 0, 00) and (0, 1, 00)) are confused at receiver 2, so S 2 must assign different colors, and S 1 must assign the same color (by Lemma 2). Thus we assign say (0, 0, 00) → (RED, RED) and (0, 1, 00) → (RED, BLUE). We color similarly for any individual sub-graph of this type. By carrying this way of coloring (as of the sub-block and the sub-graph) to all the vertices of the block with k = 1, altogether, we have the following: (0, 0, 00) → (RED, RED), (1, 0, 00) → (BLUE, RED), (0, 1, 00) → (RED, BLUE), and (1, 1, 00) → (BLUE, BLUE). We say a two-sender graph coloring is the best possible coloring if it corresponds to the minimum sum of the bits, which is required to uniquely index the colors associated with each sender, in TSUIC. As |V (D 1 )| = 1, Γ 1 (D 1 ) is a graph with two vertices (labeled by 0 and 1) connected by an edge. Thus χ (Γ 1 (D 1 )) = 2. Similarly, we get χ (Γ 1 (D 2 )) = 2. Now considering Propositions 1, 2, 3, 4, and our proposed grouping of the vertices of a confusion graph, one can get χ (Γ 1 (D 1 )) × χ (Γ 1 (D 2 )) ordered pairs of colors in the best possible coloring of a block of Γ 1 (D) in two-sender graph coloring. Thus, in any block, the four ordered pairs of colors (two colors associated per sender), that we have assigned to the vertices of Γ 1 (D), is the best possible coloring. Now we address the inter-block edges while coloring. We consider any two blocks, and perform two-sender graph coloring. Firstly, consider the two blocks with k = 1 (all tuples having b k D 3 = 00) and k = 2 (all tuples having b k D 3 = 10). As receivers 3 does not have {x 1 , x 2 } in its side-information, any tuple of the block with k = 1 and any tuple of the block with k = 2 are confused at receiver 3. Thus every vertex of the block with k = 1 are connected to each vertex of the block with k = 2 by an inter-block edge. Consequently, we do not need to consider other inter-block edges due to confusions at some other receivers for this case. Now during two-sender graph coloring of these two blocks, we need to have two different ordered pairs of colors (one for each block). Furthermore, as x 3 is a common message to both senders, it suffices to have two completely different color sets (each for one block) associated with one of the senders. In other words, one of the senders will contribute additional colors to resolve these confusions (indicated by the inter-block edges between the blocks with k equal to 1 and 2). Furthermore, for this case, one can see that if both senders contribute additional colors to resolve the confusion referring to the inter-block edges between the blocks, then the total sum of the bits that are required to represent the colors associated with S 1 and S 2 is strictly greater than three, which is required if only one sender contributes additional colors to resolve the confusion. Assume that S 1 contributes the additional colors to resolve the confusions (inter-block edges) between the vertices of these blocks. Now we have the following coloring for the vertices of the block with k = 2:  (GREEN, BLUE). Secondly, we color similarly as above for the blocks with k equal to (i) 1 and 3, (ii) 2 and 4, and (iii) 3 and 4. Thirdly, we consider the two blocks with k = 1 (all tuples having b k D 3 = 00) and k = 4 (all tuples having b k D 3 = 11). Clearly, these blocks have no inter-block edges due to the confusion at receivers 3 and 4. The inter-block edges are due to the confusion at receivers 1 and 2, and one can see them as shown in Fig. 10. It is not difficult to verify that if we color the vertices of the block with k = 4 by the same coloring function done for the vertices of the block with k = 1, which is a function of (b i D 1 , b j D 2 ) sub-labels of the vertices, then the coloring is still valid. Thus in order to color the vertices of the block with k = 4, we do not need any additional colors for senders than that assigned to the vertices of the block with k = 1. Finally, we color similarly as above for the blocks with k equal to 2 and 3. Now observe Γ 1 (D) by assuming each block as one super-vertex, the edges that connects all the vertices of one block to every vertex of another block, and vice-versa (edges due to confusion at some receivers in V (D 3 )) as a single super-edge connecting those two super-vertices, and neglect all the inter-block edges due to the confusion at some receivers except receivers in V (D 3 ), we see that the resulting graph is Γ 1 (D 3 ). Clearly, in two-sender graph coloring, we require χ (Γ 1 (D 3 )) times of χ (Γ 1 (D 1 )) × χ (Γ 1 (D 2 )) (which is required for each block) ordered pairs of colors in total, i.e., 2 × 4 = 8 ordered pairs of colors. In this way, we color the whole Γ 1 (D) with J 1 = {RED, BLUE, YELLOW, GREEN} and J 2 = {RED, BLUE}, and one can verify that this is the best possible coloring in two-sender graph coloring. Consequently, S 1 and S 2 need to transmit 2-bit and 1-bit sub-codewords, respectively. Thus β t=1 (D, G o ) = 3, and one can

B. Proof of Proposition 6
We prove the following lemmas that is used to prove the proposition. Refer to Section IV for notations. For any D, refer to Definition 9 for the definition of intra-block and inter-block edges of Γ t (D).

Lemma 7:
In two-sender graph coloring of any k-th block of the confusion graph Γ t (D) of any digraph D, the minimum number of colors associated with S 1 and S 2 are χ (Γ t (D 1 )) and χ (Γ t (D 2 )), respectively, and the minimum of χ (Γ t (D 1 )) × χ (Γ t (D 2 )) ordered pairs of colors are required to color the block.
Proof: For any D, based on our proposed way of grouping the vertices of Γ t (D) (see Section VI), we write the vertices of any k-th block of Γ t (D) in the following matrix form (for notations see Section IV): We have the following observations in B k and Γ t (D): 1) For any i, j, the sets of all the vertices of the j-th row and all the vertices of the i-th , respectively.
2) All the vertices of this block have the same b k D 3 sub-label, so we do not need to consider b k D 3 sub-label of the vertices while analyzing any k-th block of Γ t (D).

3) A sub-block (a sub-graph of Γ t (D) induced by the vertices in
sub-labels (see any row of B k ). So, we consider only b i D 1 sub-label while analyzing any j-th sub-block of Γ t (D). Now assume that we relabel the vertices of the j-th sub-block by the first sub-label, i.e., . Consequently all the rows of B k become identical. Thus we can consider one sub-block, analyze it and find its results, and these results carry similarly over all the sub-blocks of the k-th block. , and all the columns of B k become identical if we relabel the vertices of the i-th column by the second sub-label, i.e., (Necessity) For any vertex with one can think about two sets that include it, viz., (the set of vertices of the j-th row of B k ) and (the set of vertices of the i-th column of B k ) in such a way that is isomorphic to Γ t (D 1 ) (by Proposition 1), and the vertex-induced sub-graph, which is also known as the j-th sub-block of the k-th block, requires the minimum of χ (Γ t (D 1 )) different colors associated with S 1 and one color associated with S 2 , i.e., (the minimum of) χ (Γ t (D 1 )) ordered pairs of colors while coloring the sub-block in two-sender graph coloring (by Proposition 3). Thus, there exist a minimum whose colors are all different. Now considering each of these distinct vertices, one can get sets such that each set includes one of these distinct vertices. Clearly, the sub-graph of Γ t (D) induced by the vertices in the union of these disjoint sets require at least χ (Γ t (D 1 )) times the number of ordered pairs of colors required to color one of the vertex-induced sub-graph by the vertices in is isomorphic to Γ t (D 2 ) (by Proposition 2), and the vertex-induced sub-graph requires the minimum of χ (Γ t (D 2 )) different colors associated with S 2 and one color associated with S 1 , i.e., (the minimum of) χ (Γ t (D 2 )) ordered pairs of colors while coloring the sub-block in two-sender graph coloring (by Propositions 4). Altogether, the block requires at least χ (Γ t (D 1 )) and χ (Γ t (D 2 )) colors associated with S 1 and S 2 , respectively, and at least χ (Γ t (D 1 )) × χ (Γ t (D 2 )) ordered pairs of colors in its two-sender graph coloring. (i.e., vertices in the j-th row of B k ). As a sub-block requires the minimum of χ (Γ t (D 1 )) ordered pairs of colors while coloring it in two-sender graph coloring (by Proposition 3), the sub-block must have a minimum of χ (Γ t (D 1 )) independent vertex sets (one unique ordered pair of colors is assigned to one independent vertex set), otherwise we can color the sub-block with fewer than χ (Γ t (D 1 )) ordered pairs of colors, clearly this is not possible. Now for the j-th sub-block, let the sequence of independent vertex sets of this sub-block be V j,1 , V j,2 , . . . , V j, χ (Γt(D 1 )) . Considering observation (3) above, we assign the similar sequence of independent vertex sets to all other sub-blocks such that for any pair of j 1 -th and j 2 -th sub-blocks of the k-th block, any two vertex sets V j 1 ,m and V j 2 ,m for m ∈ {1, 2, . . . , χ (Γ t (D 1 ))} have the same set of vertices labeled by the first sub-label (b i D 1 ). Now writing the independent vertex sets of the block in a matrix form, we get the following matrix of order 2 tn 2 × χ (Γ t (D 1 )): where a row corresponds to the independent vertex sets of the respective sub-block, and each column of the matrix has the vertex sets having the same set of vertices labeled by the first sub-label (b i D 1 ). Now for any column m of V k , as the vertex sets of the column have the same set of vertices labeled by the first sub-label (b i D 1 ), we collect the vertices for a fixed index i to a set is isomorphic to Γ t (D 2 ), and this vertex-induced sub-graph requires the minimum of χ (Γ t (D 2 )) different colors associated with S 2 and one color associated with S 1 , i.e., (the minimum of) χ (Γ t (D 2 )) ordered pairs of colors while coloring the vertex-induced sub-graph in two-sender graph coloring (refer to Propositions 2 and 4). Thus the vertex-induced sub-graph has a minimum of χ (Γ t (D 2 )) independent vertex sets.
Now if a vertex in V j 1 ,m and a vertex in V j 2 ,m are independent, then all the vertices in V j 1 ,m and a vertex in V j 2 ,m are independent. This is because the vertex sets of the column have the same set of independent vertices labeled by the first sub-label (b i D 1 ). Thereby, for each column m, we get (the minimum of) χ (Γ t (D 2 )) independent vertex sets, and let each of these sets be denoted by V m ,m for m ∈ {1, 2, . . . , χ (Γ t (D 2 ))}. Now writing the independent vertex sets of a block in a matrix form, we get the following matrix of order χ (Γ t (D 2 )) × χ (Γ t (D 1 )): where each element of the matrix corresponds to the independent vertex sets among the respective rows and columns of V k .
In two-sender graph coloring of the k-th block, we assign one ordered pairs of colors to each of the elements in V k , so we get a χ (Γ t (D 1 )) × χ (Γ t (D 2 )) ordered pairs of colors. Let the ordered pairs of colors arranged in a matrix form be is the color of all vertices in V k (m , m) = V m ,m , for all elements of the matrices. Referring to the Definition 8 (two-sender graph coloring), one can find that this is a valid coloring, where the colors associated with S 1 and S 2 are χ (Γ t (D 1 )) and χ (Γ t (D 2 )),

respectively.
Lemma 8: Consider a two-sender graph coloring function J o that properly colors the confusion graph Γ t (D). If there is no inter-block edge due to the confusion at some receiver in , for all i and j, is a valid two-sender graph coloring. Proof: We first prove the lemma considering any two blocks, say k 1 -th block and k 2 -th block.
If there is no inter-block edge due to the confusion at any receiver in V (D 3 ) between the k 1 -th and k 2 -th blocks of Γ t (D), then we have two cases: (i) no inter-block edge and (ii) some inter-block edges due to the confusion at some receivers in V (D 1 ) ∪ V (D 2 ). In case (i), since k 1 -block and k 2 -block are isomorphic, we can color a block by two-sender graph coloring, and keep the same copy of colorings in another block (i.e., ). We know that k 1 and k 2 are two different blocks, so k 1 = k 2 . Moreover, since the edge is due to the confusion at some receivers in V (D 1 ) ∪ V (D 2 ), we must have (i, j) = (i 2 , j 2 ).
, then there must exist an edge ). This edge is between the vertices of the same block, and the confusion must have already resolved by the coloring J o . Thereby, , ∀i, j, is a valid coloring.
Since the choice of k 1 and k 2 is arbitrary, a two-sender graph coloring of all the blocks of , for all i and j, is a valid two-sender graph coloring.
Lemma 9: For D having fully-participated interactions between its sub-digraphs D 1 , D 2 and D 3 , the confusion at some receivers in V (D 1 ) does not contribute any inter-block edges in D (equivalently, (1, 3) ∈ A(f (D))), and the confusion at some receivers in V (D 2 ) does not contribute any inter-block edges in Γ t (D) if D 2 → D 3 in D (equivalently, Proof: There is no edge due to the confusion at some receivers in V (D 1 ) between any because any vertex in V (D 1 ) has {x u : u ∈ V (D 3 )} in its sideinformation and the corresponding b k 1 D 3 and b k 2 D 3 labels of the two vertices are different. This proves the first assertion. Repeating the same argument for D 2 , we get the second assertion (for the case D 2 → D 3 in D). Proof: As we know that for any real number A, we have A ≤ A , and A − A < 1 (this implies A < A + 1 or A − 1 < A). So we get A ≤ A < A + 1. This is true for any other real number A + B, so A + B ≤ A + B < A + B + 1. Altogether, we get , then each of the vertices in the k 1 -th block has edges with all the vertices in the k 2 -th block. This is because any vertex in V (D 3 ) has no message requested by any vertex in V (D 1 ) ∪ V (D 2 ) in its side-information resulting in there being no effect due to a change in bits of b i D 1 or b j D 2 sub-label once we have an edge due to confusion at some receivers in V (D 3 ) that corresponds to the change in bits of the b k D 3 sub-label. (Coloring the confusion graph Γ t (D 27 )) In SSUIC, we know that the minimum numbers of colors required to color D 1 , D 2 and D 3 separately are χ (Γ t (D 1 )), χ (Γ t (D 2 )) and χ (Γ t (D 3 )), respectively. From Lemma 7, in two-sender graph coloring, vertices in any k-th block of Γ t (D 27 ) are colored properly with the minimum of χ (Γ t (D 1 )) × χ (Γ t (D 2 )) ordered pairs of colors, where the minimum number of colors associated with S 1 and S 2 are χ (Γ t (D 1 )) and χ (Γ t (D 2 )), respectively. Referring to the construction of Γ t (D 27 ), the inter-block edges are solely due to the confusion at some vertices in V (D 3 ) (from (i), (ii) and (iii) of the construction), and if there exists an inter-block edge between any two vertices, the first one belonging to k 1 -th block and the second one belonging to k 2 -th block, then we have edges from every vertex of the k 1 -th block to all vertices of the k 2 -th block. This states that it is necessary to have two different sets of ordered pairs of χ (Γ t (D 1 )) × χ (Γ t (D 2 )) colors, one for each block if there is an edge between these blocks. Furthermore, it is sufficient to consider the different color sets associated with one of the senders for those blocks in order to obtain the different sets of χ (Γ t (D 1 )) × χ (Γ t (D 2 )) ordered pairs of colors. As we require the minimum of χ (Γ t (D 3 )) ordered pairs of colors to color vertices in any (refer to Propositions 5), so the total number of minimum ordered pairs of colors required to color Γ t (D 27 ) in two-sender graph ) and χ 2 (Γ t (D 3 )) be the nonnegative non-zero integer factors of χ (Γ t (D 3 )) which are the best choice over all χ 1 (Γ t (D 3 )) and is minimized. Here an integer is considered because each sender broadcasts sub-codewords in bits. Now χ (Γ t (D 1 )) × χ 1 (Γ t (D 3 )) and χ (Γ t (D 2 )) × χ 2 (Γ t (D 3 )) are the colors associated with S 1 and S 2 , respectively, and they produce χ (Γ t (D 1 )) × χ (Γ t (D 2 )) × χ (Γ t (D 3 )) ordered pairs of colors required for two-sender graph coloring of Γ t (D 27 ). Now from Theorem 2, we get where 1 ∈ {−2, −1, 0} and 2 ∈ {0, 1} are obtained by using Lemma 10, and = ( 2 (13)), the value of in (29) cannot be greater than zero. Thus, For the extra inter-block edges in E(Γ t (D 1 )) due to the confusion at some vertices in V (D 1 ) and V (D 2 ) in two-sender graph coloring, we have the following: If there is no inter-block edge due to the confusion at some vertices in V (D 3 ), then we can do two-sender graph coloring of these blocks as stated by Lemma 8. This implies that we can do two-sender graph coloring of all these blocks by χ (Γ t (D 1 )) × χ (Γ t (D 2 )) ordered pairs of colors, where the minimum colors associated with S 1 and S 2 are χ (Γ t (D 1 )) and χ (Γ t (D 2 )), respectively. Now referring to the edges in E(Γ t (D 1 )) due to the confusion at some vertices in V (D 3 ), if any k 1 -th and k 2 -th blocks have the inter-block edges (includes all extra edges due to the confusion at some vertices in V (D 1 ) and V (D 2 )), then it is necessary to have two different sets of ordered pairs of χ (Γ t (D 1 )) × χ (Γ t (D 2 )) colors, one for each block (the same as of the case of D 27 ). Furthermore, it is sufficient to consider different color sets associated with one of the senders for these blocks in order to achieve the necessity.
Altogether, similar to D 27 , the total number of minimum ordered pairs of colors required to color Γ t (D 1 ) in two-sender graph coloring is χ (Γ t (D 1 )) × χ (Γ t (D 2 )) × χ (Γ t (D 3 )). Thus the results for APPENDIX C

PROOF OF THEOREM 7
Proof: For the problems in TSUIC, we prove this theorem by constructing a valid index code based on single-sender index codes. Refer to Section IV for notations.
for a finite t. Now in TSUIC, we propose that S 1 transmits )+p(C(D 1 )) and p 1 (C(D 3 )) = p(C(D 1 )). Each receiver receives (p 1 (C(D 3 )) + p 2 (C(D 3 )))-bit (C 1 , C 2 ). Now the decoding is done in the following way: (i) All the vertices in V (D 1 ) will decode their requested messages from C 1 and its side-information that also includes {x i : i ∈ V (D 3 )} (as there is a fully-participated D 1 → D 3 in D), (ii) all the vertices in V (D 2 ) will decode their requested messages from C 2 and its side-information that also includes {x i : i ∈ V (D 3 )} (as there is a fully-participated D 2 → D 3 in D), and (iii) all the vertices in V (D 3 ) will decode their requested messages from (C 1 , C 2 ) and its side-information that also includes {x i : i ∈ V (D 1 ) ∪ V (D 2 )} (as there is fully-participated D 3 → (D 1 ∪ D 2 ) in D). Thus (p 1 (C(D 3 )) + p 2 (C(D 3 )))-bit (C 1 , C 2 ) is a valid index code in TSUIC for this case, and β t (D, G o ) ≤ (p 1 (C(D 3 )) + p 2 (C(D 3 )))/t = p(C(D 3 ))/t = β t (D 3 ).
Each receiver receives (p(C(D 1 )) + p(C(D 1 )))-bit (C 1 , C 2 ). Now one can verify that the decoding is done in the same way as stated in the first case. Thus (p(C(D 1 )) + p(C(D 1 )))-bit (C 1 , C 2 ) is a valid index code in TSUIC for this sub-case.
(Sub-case (ii): |C(D 3 )| ≤ |C(D 1 )|) S 1 transmits C 1 = C(D 3 )⊕C(D 1 ) of p(C(D 1 )) bits (because p(C(D 3 )) ≤ p(C(D 1 ))), and S 2 transmits C 2 = C(D 2 ) of p(C(D 1 )) bits. Now the decoding is done in the following way: (i) All the vertices in V (D 1 ) will decode their requested messages from C 1 and its side-information that also includes {x i : i ∈ V (D 3 )} (as there is a fully-participated , (ii) all the vertices in V (D 2 ) will decode their requested messages from C 2 and its side-information, and (iii) all the vertices in V (D 3 ) will decode their requested messages from C 1 and its side-information that also includes {x i : i ∈ V (D 1 )} (as there is a fully-participated D 3 → D 1 in D). Thus (p(C(D 1 )) + p(C(D 1 )))-bit (C 1 , C 2 ) is a valid encoding in TSUIC for this sub-case.
(Sub-case (iii): |C(D 3 )| ≤ |C(D 2 )|) As we have a fully-participated D 3 → (D 1 ∪ D 2 ) in D, so by swapping D 1 and D 2 (meaning we swap the two senders) in the sub-case (ii), one can prove that (p(C(D 1 )) + p(C(D 1 )))-bit (C 1 , C 2 ) is a valid index code in TSUIC for this sub-case.
By considering β t (D, G o ) ≥ β t (D 1 ) + β t (D 2 ) (by Lemma 6) and (32), we get, β t (D, G o ) = β t (D 1 ) + β t (D 2 ). Now combining these two cases (First and Second cases), we get Now taking a limit t → ∞ on both sides of (33), we get Proof: For the problems in TSUIC, we prove this theorem by constructing a valid index code based on single-sender index codes. Refer to Section IV for notations.
In TSUIC, we propose that S 1 transmits C 1 = C(D 1 ) ⊕ C(D 3 ) consisting of max{p(C(D 1 )), p(C(D 3 ))} bits, and S 2 transmits C 2 = C(D 2 ) of p(C(D 1 )) bits. Each receiver receives (p(C(D 1 ))+ max{p(C(D 1 )), p(C(D 3 ))})-bit (C 1 , C 2 ). Now the decoding is done in the following way: (i) All the vertices in V (D 1 ) will decode their requested messages from C 1 and its side-information that also includes {x i : i ∈ V (D 3 )} (as there is a fully-participated D 1 → D 3 in D), (ii) all the vertices in V (D 2 ) will decode their requested messages from C 2 and its side-information, and (iii) all the vertices in V (D 3 ) will decode their requested messages from C 1 and its side-information that also includes {x i : i ∈ V (D 1 )} (as there is a fully-participated D 3 → D 1 in D). Thus For the sub-digraph D j , j ∈ {1, 2, 3}, we know that there exists an index code C(D j ) of p(C(D j )) bits such that p(C(D j ))/t tends to β(D j ) if t → ∞, and for any t ≥ 1, β(D j ) ≤ p(C(D j ))/t. So, for we write p(C(D j ))/t = β(D j ) + t (D j ), for some t (D j ) ≥ 0 such that t (D j ) tends to zero if message length t tends to infinity. For D, considering the same code formation, which is a valid two-sender index code, as stated for the cases considering the finite message length, we get S 1 and S 2 transmitting sub-codewords of max{p(C(D 1 )), p(C(D 3 ))} and p(C(D 2 )) bits, respectively. For any t ≥ 1, there exists a two-sender index code of the following bit length: p 1 + p 2 = max{p(C(D 1 )), p(C(D 3 ))} + p(C(D 2 )).
Alternatively, we can get (37) by taking a limit t → ∞ on both sides of (35) because β = lim

APPENDIX E PROOF OF THEOREM 9
Proof: For the problems in TSUIC, we prove this theorem by constructing a valid index code based on single-sender index codes. Refer to Section IV for notations.
In TSUIC, we propose that S 1 transmits will decode their requested messages from C 1 ⊕ C 2 and its side-information that also includes