Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs 2,1,2

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring φ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u} φ(z) 6= ∑z∈EG(v)∪{v} φ(z) for each edge uv ∈ E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree ∆ admits an NSD total (∆ + 3)coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree ∆ ≥ 9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.


Introduction
The graphs mentioned in this paper are finite, undirected, and simple. For undefined terminology and notations, here we follow [1]. Let G = (V(G), E(G)) be a simple graph. For a vertex u ∈ V(G), let E G (u) denote the set of edges incident with u, and we use d G (u) and N G (u) to represent the degree and the neighborhood of u, respectively. Let ∆(G) (or ∆) and δ(G) (or δ) denote the maximum degree and the minimum degree of G, respectively. A t-cycle (t + -cycle, t − -cycle) is a cycle of length t (at least t, at most t). In particular, a 3-cycle with vertex set {v 1 , v 2 , v 3 } is called a (d G (v 1 ), d G (v 2 ), d G (v 3 ))-cycle. A theta graph Θ t 1 ,t 2 ,t 3 is a graph obtained by joining two vertices by three internally disjoint paths of lengths t 1 , t 2 and t 3 .
Let k be a positive integer and T(G) = V(G) ∪ E(G). A mapping φ : T(G) → {1, · · · , k} is called a proper k-total coloring of G if φ(z 1 ) = φ(z 2 ) for any two adjacent or incident elements z 1 , z 2 in T(G). A proper k-total coloring φ of G is neighbor sum distinguishing (for short, NSD) if for each edge uv ∈ E(G), The NSD total chromatic number of G, denoted by χ t Σ (G), is the smallest integer k such that G has an NSD k-total coloring. Pilśniak and Woźniak [2] posed an important conjecture in the following.
An IC-planar graph, introduced by Alberson [6] in 2008, is a graph that can be embedded in a plane such that each edge is crossed at most one other edge and two pairs of crossing edges share no common end vertex, i.e., two distinct crossings are independent. There are also many results about the NSD total coloring of IC-planar as follows.
A k-list total assignment of G is a mapping L that assigns to each member z ∈ T(G) a set L(z) of k real numbers. For a list total assignment L of G, a mapping φ is called an NSD total L-coloring of G if φ is an NSD total coloring of G and φ(z) ∈ L(z) for each z ∈ T(G). The smallest integer k such that G has an NSD total L-coloring for any k-list total assignment L is called the NSD total choice number of G, denoted by ch t Σ (G). Clearly, χ t Σ (G) ≤ ch t Σ (G). There are also many results about the list version of Conjecture 1 in the following.
In this paper, we reduce the condition ∆ ≥ 10 of (3) in Theorem 1 to ∆ ≥ 9 and obtain the list version result as follows.

Preliminaries
Let G be a simple graph. An -vertex ( . We use n G (u) (n + G (u), n − G (u)) to denote the number of -vertices ( + -vertices, − -vertices) adjacent to u.
In 1999, Alon developed a general algebraic technique that is called Combinatorial Nullstellensatz. It has numerous applications in additive number theory, combinatorics, and graph coloring problems. In the paper, we also use Combinatorial Nullstellensatz in the following to discuss the local structure of minimal counterexample to Theorem 3.
Lemma 1 ( [13]). Let F be an arbitrary field and P ∈ F[x 1 , . . . , x n ] with degree deg(P) = ∑ n k=1 i k , where each i k is a natural integer. If the coefficient c P (x i 1 1 · · · x i n n ) of the monomial x i 1 1 . . . x i n n in P is nonzero, and if S 1 , . . . , S n are subsets of F with |S k | > i k , then there are s 1 ∈ S 1 , . . . , s n ∈ S n such that P(s 1 , . . . , s n ) = 0.
Let t ≥ 2 be a positive integer and S 1 , · · · , S t be t finite sets of real numbers. Define s i | s i ∈ S i and s i = s j f or 1 ≤ i < j ≤ t}.

Proof of Theorem 3
Let G be a counterexample to Theorem 3 with |E(G)| being minimal and k = max{∆(G) + 3, 12}. For any k-list total assignment L, every subgraph G of G has an NSD total L-coloring φ by the minimality of G. In the following, we will extend the NSD total L-coloring φ of G to an NSD total L-coloring φ of G to obtain a contradiction. As G is a subgraph of G, Assume that u is a 3 − -vertex of G. For any k-list total assignment L of G, let a map φ : T(G) \ {u} → R satisfy the following three conditions: As |S(u)| ≥ k − 2d G (u) > d G (u) + 1 follows from the assumption that k ≥ 12 and d G (u) ≤ 3, there is a color in S(u) to color u such that the resulting coloring φ obtained from φ satisfies m(u) = m (z) for each z ∈ N G (u) and φ(u) = φ (z) for each z ∈ N G (u) ∪ E G (u). Therefore, φ satisfies the definition of NSD total L-coloring. Therefore, φ can be extend to an NSD total L-coloring φ of G. Thus, for simplicity, we will omit the colors of all 3 − -vertices in the following. Theorem 3 follows from Theorem 2 if ∆(G) > 13. Thus, the following Claim 1 is immediate.

Proof. By contradiction, suppose that there is an edge
Then, G has an NSD total L-coloring φ . In order to extend the coloring φ to an NSD total L-coloring φ of G, we erase the colors on u and v. Then, Assign a variable x 1 to u, a variable y 1 to uv and a variable x 2 to v, respectively. Let . By Appendix A, we know that c P (x 3 1 y 4 1 x 4 2 ) = 5. By Lemma 1, there are s 1 ∈ S(u), s 2 ∈ S(uv) and s 3 ∈ S(v) such that P(s 1 , s 2 , s 3 ) = 0. By the definitions of P and NSD total L-coloring, we can extend φ to an NSD total L-coloring φ of G by recoloring u and v with colors s 1 , s 3 and coloring uv with color s 2 . It is a contradiction.
Proof. Suppose to be contrary that there is an edge uv in G with d G (u) ≤ 6 and d G (v) ≤ 2. Without loss of generality, set d G (u) = 6. Let G = G − uv. Then, G has an NSD total L-coloring φ . In order to extend the coloring φ to an NSD total L-coloring φ of G, we erase the colors on u and v. Note that v is a 2 − -vertex. The color of v can be omitted when we extend φ to an NSD total L-coloring φ of G. Then, By Lemma 2, we have that Thus, there is a color in S(u) to color u and a color in S(uv) to color uv such that the resulting color φ obtain from φ satisfies m(u) = m (z) for each z ∈ N G (u) \ {v}. Therefore, we can extend φ to an NSD total L-coloring φ of G. It is a contradiction.
The proofs of the following Claims 4 and 5 are similar to the proof of Claim 2. To avoid duplication, we omit the proofs. The proof of the following Claim 6 is similar to the proof of Claim 3. To avoid duplication, we omit the proof of Claim 6. Claim 6. Let u be an -vertex of G. Then, each of the following results must hold.

Claim 7.
For the graph H, each of the following results must hold.
From now on, we assume that the IC-planar graph G has been embedded on a plane such that every edge is crossed by at most one other edge and the number of crossings is as small as possible. We turn all crossings of G into new 4-vertices on the plane and obtain a planar graph G × , which is called the associated planar graph of G. A vertex in G × is called a false vertex if it is not a vertex of G and real vertex otherwise. We call a face f in G × a false face if it is incident with one false vertex and a real face otherwise.
Let H × be the associated planar graph of H. For each real vertex v ∈ V(H × ), we use f (v) and f (v) to denote the number of real 3-faces and false 3-faces incident with v, respectively.

Claim 9.
Let v be a real vertex of H × . Then, each of the following results must hold. ( . ( ( . Proof. (1) As G (and thus H) is an IC-planar graph without theta graphs Θ 2,1,2 , any two real 3-faces have no common edge in H. Thus, a real vertex v of H × is incident with at most d H × (v) 2 real 3-faces as each 3-face contains two edges incident with v. Furthermore, statement (1) holds.
(2) By contradiction, suppose that . Thus, each edge incident with v belongs to a real 3-face incident with v since each 3-face contains two edges. Moreover, v is not adjacent to any false (4) Let v be incident with two false 3-faces. Then, the two false 3-faces contain three edges incident with v since G (and thus H) is an IC-planar graph without theta graphs Θ 2,1,2 . Thus, the two false 3-faces cause that f (v) is reduced by 1 when d H × (v) ≡ 1 (mod 2) and 2 when d H × (v) ≡ 0 (mod 2). Therefore, (3) and (4) hold.
The discharging method, first developed in the study of the coloring of planar graphs about 100 years ago, is an important proof technique in graph theory. The method has been applied in many types of problems, especially in various graph coloring problems.
The general process of discharging is that members (usually vertices or vertices and faces) of a graph are assigned charges by certain "charging rules", then the graph is discharged by certain "discharging rules", during which some members get charges, and some members lose charges, while the sum of the charges keeps unchanged.
In the following, we will apply the discharging method on the associated planar graph H × to show that H × (and thus H) does not exist. Therefore, G does not exist.
Let Next, we make some discharging rules to redistribute charges among vertices and faces and keep the total charges unchanged. For simplicity, a real -vertex is still called an -vertex in the following discussion. The discharging rules as follows: In the following, we give a specific example about the charge change of some false 4vertex by the discharging rules. Let u be a false 4-vertex in H × and Figure 1) as a specific example to illustrate how the charge of the false 4-vertex u changes by the discharging rules. Next, we discuss the new charge of each z ∈ V(H × ) ∪ F(H × ) after the discharging process. Let ω (z) denote the new charge for each z ∈ V(H × ) ∪ F(H × ). Then, In the following, we show ω (z) ≥ 0 for each z ∈ V(H × ) ∪ F(H × ) to obtain a contradiction.
Note that as H is an IC-planar graph, each real vertex z is adjacent to at most a false 4-vertex and 0 ≤ f (z) ≤ 2.