Strong Chromatic Index of Outerplanar Graphs

Abstract

The strong chromatic index ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)$ of a graph G is the minimum number of colors needed in a proper edge-coloring so that every color class induces a matching in G. It was proved In 2013, that every outerplanar graph G with $\Delta \ge 3$ has ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 3\Delta -3$. In this paper, we give a characterization for an outerplanar graph G to have ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=3\Delta -3$. We also show that if G is a bipartite outerplanar graph, then ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 2\Delta$; and ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=2\Delta$ if and only if G contains a particular subgraph.

1. Introduction

Only simple graphs are considered in this paper. For a graph G, we use $V\left(G\right)$, $E\left(G\right)$, and $\Delta \left(G\right)$ to denote its vertex set, edge set and maximum degree, respectively. A vertex v is called a k-vertex (or $k^{+}$-vertex) if the degree $d_G\left(v\right)$ of v is k (or at least k). Let $N_G\left(v\right)$ denote the set of vertices adjacent to v in G. If no ambiguity arises in the context, $\Delta \left(G\right)$, $d_G\left(v\right)$, and $N_G\left(v\right)$ are simply written as $\Delta$, $d\left(v\right)$, and $N\left(v\right)$, respectively. A subgraph of G is called a clique if any two of its vertices are adjacent in G. A subset $I\subset V\left(G\right)$ of a connected graph G is called a clique-cut if $G\left[I\right]$ is a clique and $G-I$ is disconnected.

A proper edge-k-coloring of a graph G is a mapping $\varphi :E\left(G\right)\to \{1,2,\dots ,k\}$ such that $\varphi \left(e\right)\ne \varphi \left(e^{\prime }\right)$ for any two adjacent edges e and $e^{\prime }$. The chromatic index${\chi }^{\prime }\left(G\right)$ of G is the smallest k such that G has a proper edge k-coloring. An edge coloring of the graph G is called strong if every color class induces a matching in G. The strong chromatic index of G, denoted ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)$, is the smallest k such that G has a strong edge-k-coloring.

The strong edge-coloring of graphs was introduced by Fouquet and Jolivet [1]. In 1985, Erdos and Nešetřil raised the following conjecture and showed that the upper bounds are tight:

Conjecture 1.

For a graph G,

$${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le \left\{\begin{array}{cc}1.25{\Delta }^2,\hfill & \mathrm{if}\Delta \mathrm{is}\mathrm{even};\hfill \\ 1.25{\Delta }^2-0.5\Delta +0.25,\hfill & \mathrm{if}\Delta \mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

Using probabilistic method, Molloy and Reed [2] showed that ${\chi }_s^{\prime }\left(G\right)\le 1.998{\Delta }^2$ when $\Delta$ is sufficiently large. This result was further improved in [3] to that ${\chi }_s^{\prime }\left(G\right)\le 1.93{\Delta }^2$ for any graph G. Using Four-Colour Theorem and Vizing Theorem, Faudree et al. [4] showed that every planar graph G has ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 4\Delta +4$; and constructed a planar graph G such that ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=4\Delta -4$.

A planar graph is called outerplanar if it has a plane embedding such that all the vertices lie on the boundary of the unbounded face. It was shown in [5] that a graph G is outerplanar if and only if G is $K_4$-minor-free and $K_{2,3}$-minor-free. Hence outerplanar graphs are special $K_4$-minor-free graphs. Wang et al. [6] showed that every $K_4$-minor-free graph G with $\Delta \ge 3$ has ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 3\Delta -2$ and the upper bound is tight. Hocquard et al. [7] proved that every outerplanar graph G with $\Delta \ge 3$ has ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 3\Delta -3$ and the upper bound is tight.

In this paper we will give a characterization for an outerplanar graph G with $\Delta \ge 3$ to have ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=3\Delta -3$.

2. Sun-Graphs

Suppose that G is an outerplanar graph. We embed G in the plane so that all the vertices occur in the boundary of unbounded face. Let $F\left(G\right)$ denote the set of faces in G. The unbounded face, denoted by $f_0\left(G\right)$, of G is called outer face, and other faces inner faces. For a face $f\in F\left(G\right)$, the boundary of f is denoted by $b\left(f\right)$. A 3-face with $x,y,z$ as boundary vertices is written as $\left[xyz\right]$. The edges lying in the outer face are called outer edges and other edges inner edges. An inner face f is called an end-face if $b\left(f\right)$ contains at most one inner edge. A leaf of G is a vertex of degree 1, and a pendant edge is an edge incident to a leaf. For a vertex $v\in V\left(G\right)$, let $L\left(v\right)$ denote the set of pendant edges at vertex v. For a cycle C, an edge $xy\in E\left(G\right)\backslash E\left(C\right)$ is called a chord of C if $x,y\in V\left(C\right)$.

Let $F_1$ denote a subgraph of G, which consists of a 3-cycle $C_3=x_0{x}_1{x}_2{x}_0$ with $d_G\left(x_i\right)=\Delta \ge 3$ for $i=0,1,2$.

Let $F_2$ denote a subgraph of G, which consists of a 4-cycle $C_4=x_0{x}_1{x}_2{x}_3{x}_0$ with $d_G\left(x_0\right)=d_G\left(x_1\right)=\Delta \ge 3$.

Let $F_3$ denote a subgraph of G, which consists of a 7-cycle $C_7=x_0{x}_1\cdots x_6{x}_0$ with $d_G\left(x_i\right)=3$ for $i=0,1,\dots ,6$.

We assume that $C_4$ in $F_2$ and $C_7$ in $F_3$ have no chord.

The configurations $F_1,F_2,F_3$ are depicted in Figure 1. By the outerplanarity of G, for $F_j$, $j\in \{1,2,3\}$, some vertex $y_i\in N\left(x_i\right)\backslash \{x_{i-1},x_{i+1}\}$ may identify with some vertex $y_{i+1}\in N\left(x_{i+1}\right)\backslash \{x_i,x_{i+2}\}$, but there is at most one such pair $\{y_i,y_{i+1}\}$ satisfying $y_i=y_{i+1}$, where indices i are taken as modulo n.

Lemma 1

([7]). If G is an outerplanar graph with $\Delta \ge 3$, then ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 3\Delta -3$.

Lemma 2.

Let $F_1,F_2,F_3$ are defined as above. Then

(1)

${\chi }_{\mathrm{s}}^{\prime }\left(F_1\right)=3\Delta -3$.

(2)

${\chi }_{\mathrm{s}}^{\prime }\left(F_2\right)=3\Delta -3$.

(3)

${\chi }_{\mathrm{s}}^{\prime }\left(F_3\right)=6$.

Proof. 

(1)

Since $\left|E\right(F_1\left)\right|=3\Delta -3$ and it is easy to check that any two edges of $F_1$ have distance at most two, so it follows that ${\chi }_{\mathrm{s}}^{\prime }\left(F_1\right)=3\Delta -3$.

(2)

Applying the similar analysis as in (1), we can derive that ${\chi }_{\mathrm{s}}^{\prime }\left(F_2\right)=3\Delta -3$.

(3)

It is evident that ${\chi }_{\mathrm{s}}^{\prime }\left(F_3\right)\le 6$ by Lemma 1. Conversely, assume that $F_3$ admits a strong edge-5-coloring $\varphi$ using the color set $C=\{1,2,\dots ,5\}$. Let $E_i$ denote the set of edges colored with the color i under the coloring $\varphi$. Set $E^{*}=E\left(F_3\right)-E\left(C_7\right)$. First, it is easy to inspect that $|E_i|\le 3$ for each $i\in C$. Next, because $\left|E\right(F_3\left)\right|=14$ and $\left|C\right|=5$, we can assume that $|E_i|=3$ for $i=1,2,3,4$ and $|E_5|=2$. Since $|E^{*}|=7$, some $E_i$ for $i\in \{1,2,3,4\}$, say $i=1$, satisfies $|E_1\cap E^{*}|\le 1$. It implies that $|E_1\cap E\left(C_7\right)|\ge 2$. On the other hand, it is easy to inspect that $|E_1\cap E\left(C_7\right)|\le 2$. So $|E_1\cap E\left(C_7\right)|=2$ and $|E_1\cap E^{*}|=1$, however such coloring is impossible, a contradiction. This shows that ${\chi }_{\mathrm{s}}^{\prime }\left(F_3\right)\ge 6$. Consequently, ${\chi }_{\mathrm{s}}^{\prime }\left(F_3\right)=6$.

 □

Let $C_n=x_0{x}_1\cdots x_{n-1}{x}_0$ be a cycle with $n\ge 3$. Let $k\ge 3$ be an integer. At each vertex $x_i$, we glue $k-2$ leaves and write the resultant graph as $S_n^k$. Then $S_n^k$ is an outerplanar graph with maximum degree k and order $n(k-1)$. We call $S_n^k$ a sun-graph with parameters n and k. If $k=3$, then we use $y_i$ to denote a leaf adjacent to $x_i$ for $i=0,1,\dots ,n-1$.

As an easy observation, we have the following:

Lemma 3.

Let $C_n$ be a cycle with $n\ge 3$. Then

$${\chi }_{\mathrm{s}}^{\prime }\left(C_n\right)=\left\{\begin{array}{cc}5,\hfill & \mathrm{if}\mathrm{n}=5;\hfill \\ 3,\hfill & \mathrm{if}n\equiv 0\left(\mathrm{mod}3\right);\hfill \\ 4,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Lemma 4.

Let $S_n^3$ be a sun-graph with $n\ge 3$. Then

$${\chi }_{\mathrm{s}}^{\prime }\left(S_n^3\right)=\left\{\begin{array}{cc}6,\hfill & \mathrm{if}\mathrm{n}=3,4,7;\hfill \\ 5,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Proof. 

If $n=3,4,7$, the conclusion follows immediately from Lemma 2. So suppose that $n\ne 3,4,7$. It holds trivially that ${\chi }_{\mathrm{s}}^{\prime }\left(S_n^3\right)\ge 5$ since $S_n^3$ contains two adjacent 3-vertices. To show that ${\chi }_{\mathrm{s}}^{\prime }\left(S_n^3\right)\le 5$, we make use of induction on n. It remains to construct a strong edge-5-coloring $\varphi$ of $S_n^3$ using the color set $C=\{1,2,\dots ,5\}$.

• If $n=5$, then we color the edges in $\{x_i{y}_i,x_{i+2}{x}_{i+3}\}$ with $i+1$ for $i=0,1,2,3,4$, where indices are taken as modulo 5.
• If $n\equiv 0$ (mod 6), then we alternatively color the edges in $E\left(C_n\right)$ with $1,2,3$, and color alternatively pendant edges with $4,5$.
• If $n=8$, then we color $\{x_1{y}_1,x_3{x}_4,x_6{x}_7\}$ with 1, $\{x_3{y}_3,x_0{x}_1,x_5{x}_6,\}$ with 2, $\{x_5{y}_5,x_0{x}_7,\}$$\left\{x_2{x}_3\right\}$ with 3, $\{x_7{y}_7,x_1{x}_2,x_4{x}_5\}$ with 4, and $\{x_0{y}_0,x_2{y}_2,x_4{y}_4,x_6{y}_6\}$ with 5.
• If $n=9$, then we color $\{x_1{y}_1,x_3{y}_3,x_8{y}_8,x_5{x}_6\}$ with 1, $\{x_0{y}_0,x_5{y}_5,x_7{y}_7,x_2{x}_3\}$ with 2, $\{x_2{y}_2,x_4{y}_4,x_6{y}_6,x_0{x}_8\}$ with 3, $\{x_0{x}_1,x_3{x}_4,x_6{x}_7\}$ with 4, and $\{x_1{x}_2,x_4{x}_5,x_7{x}_8\}$ with 5.

Now assume that $n\ge 10$ and $n\neg \equiv 0$ (mod 6). Consider the graph $S_{n-5}^3$. Note that $n-5\ge 5$, and $n-5\ne 7$. By the induction hypothesis, ${\chi }_{\mathrm{s}}^{\prime }\left(S_{n-5}^3\right)=5$. Let $\varphi$ be a strong edge-5-coloring of $S_{n-5}^3$, so that $\varphi \left(x_0{x}_1\right)=1$, $\varphi \left(x_0{y}_0\right)=2$, $\varphi \left(x_{n-7}{x}_{n-6}\right)=3$, $\varphi \left(x_{n-6}{y}_{n-6}\right)=4$, and $\varphi \left(x_0{x}_{n-6}\right)=5$. Clearly, $S_n^3$ can be obtained from $S_{n-5}^3$ by inserting five vertices $x_{n-5}$, $x_{n-4}$, $x_{n-3}$, $x_{n-2}$, $x_{n-1}$ to the edge $x_0{x}_{n-6}$ and adding a leaf $y_j$ at $x_j$ for $j=n-5,n-4,\dots ,n-1$. We extend $\varphi$ to $S_n^3$ by coloring $\{x_{n-3}{x}_{n-2}$, $x_{n-5}{y}_{n-5}\}$ with 1, $\{x_{n-5}{x}_{n-4},x_{n-2}{y}_{n-2}\}$ with 2, $\{x_{n-2}{x}_{n-1},x_{n-4}{y}_{n-4}\}$ with 3, $\{x_{n-4}{x}_{n-3},x_{n-1}{y}_{n-1}\}$ with 4, and $\{x_0{x}_{n-1},x_{n-6}{x}_{n-5},x_{n-3}{y}_{n-3}\}$ with 5. It is easy to testify that the extended coloring is a strong edge-5-coloring of $S_n^3$. □

For a sun-graph $S_n^k$ with $C_n=x_0{x}_1\cdots x_{n-1}{x}_0$, we set $L\left(x_i\right)=\{e_i^1,e_i^2,\dots ,e_i^{k-2}\}$ for $i=0,1,\dots ,n-1$. Recall that $L\left(x_i\right)$ stands for the set of pendant edges incident to $x_i$.

Lemma 5.

Let $S_n^k$ be a sun-graph with $k,n\ge 4$ and n being even. Then

$${\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)=\left\{\begin{array}{cc}2k,\hfill & \mathrm{if}n=4;\hfill \\ 2k-1,\hfill & \mathrm{if}n\ge 6.\hfill \end{array}\right.$$

Proof. 

Since $k\ge 4$, it follows that $k-2\ge 2$. The proof is split into the following two cases.

• Assume that $n=4$. Color $x_0{x}_1,x_1{x}_2,x_2{x}_3,x_3{x}_0$ with $1,2,3,4$, respectively; For $i=0,2,\dots ,n-2$, we color $k-2$ pendant edges in $L\left(x_i\right)$ with colors $5,6,\dots ,k+2$; For $i=1,3,\dots ,n-1$, we color $k-2$ pendant edges in $L\left(x_i\right)$ with colors $k+3,k+4,\dots ,2k$. It is easy to see that the defining coloring is a strong edge-$2k$-coloring of $S_4^k$. Hence ${\chi }_{\mathrm{s}}^{\prime }\left(S_4^k\right)\le 2k$. Conversely, we note that every pendant edge of $S_4^k$ has distance at most two to any edge in $E\left(C_4\right)$. This implies that, for any strong edge coloring of $S_4^k$, the color of any pendant edge is distinct from that of edges in $C_4$. Moreover, at least $2(k-2)$ colors are needed when we color the $4(k-2)$ pendant edges of $S_4^k$. It follows therefore that ${\chi }_{\mathrm{s}}^{\prime }\left(S_4^k\right)\ge 4+2(k-2)=2k$. This yields that ${\chi }_{\mathrm{s}}^{\prime }\left(S_4^k\right)=2k$.
• Assume that $n\ge 6$. It is straightforward to conclude that ${\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)\ge 2k-1$ since $S_n^k$ contains two adjacent k-vertices. Conversely, we notice that $S_n^3$ is a spanning subgraph of $S_n^k$. By Lemma 4, $S_n^3$ has a strong edge-5-coloring $\varphi$ using colors $1,2,3,4,5$. Based on $\varphi$, we can color the remaining $k-3$ pendant edges in $L\left(x_i\right)$ with colors $6,7,\dots ,k+2$ for each $i=0,2,\dots ,n-2$; and color the remaining $k-3$ pendant edges in $L\left(x_i\right)$ with colors $k+3,k+4,\dots ,2k-1$ for each $i=1,3,\dots ,n-1$. The extended coloring is a strong edge-$(2k-1)$-coloring of $S_n^k$. It therefore turns out that ${\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)\le 2k-1$. Consequently, ${\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)=2k-1$.

 □

Lemma 6.

Let $n\ge 4$ be an odd number. Then

(1)

${\chi }_{\mathrm{s}}^{\prime }\left(S_n^4\right)\le 8$.

(2)

${\chi }_{\mathrm{s}}^{\prime }\left(S_n^5\right)\le 11$.

Proof. 

We first prove (1), by discussing two cases below.

• Assume that $n=7$. Give a strong edge-7-coloring $\varphi$ of $S_7^4$ as follows: $\varphi \left(x_i{x}_{i+1}\right)=i+1$ for $i=0,1,\dots ,6$, where indices are taken as modulo 7; then we color $L\left(x_0\right)$ with $3,5$, $L\left(x_1\right)$ with $4,6$, $L\left(x_2\right)$ with $5,7$, $L\left(x_3\right)$ with $1,6$, $L\left(x_4\right)$ with $2,7$, $L\left(x_5\right)$ with $1,3$, and $L\left(x_6\right)$ with $2,4$.
• Assume that $n\ne 7$. By Lemma 4, $S_n^3$ admits a strong edge-5-coloring $\varphi$ using the colors $1,2,\dots ,5$ so that $e_0^1,e_1^1,\dots ,e_{n-1}^1$ have been colored. Afterward, we extend $\varphi$ to the remaining edges of $S_n^4$ by coloring $e_0^2$ with 6, $\{e_1^2,e_3^2,\dots ,e_{n-2}^2\}$ with 7, and $\{e_2^2,e_4^2,\dots ,e_{n-1}^2\}$ with 8. It is easily seen that the resultant coloring is a strong edge-5-coloring of $S_n^4$.

Next we prove (2). By the result of (1), $S_n^4$ has a strong edge-8-coloring $\varphi$ using the colors $1,2,\dots ,8$. Based on $\varphi$, we can color $e_0^3$ with 9, $\{e_1^3,e_3^3,\dots ,e_{n-2}^3\}$ with 10, and $\{e_2^3,e_4^3,\dots ,e_{n-1}^3\}$ with 11. This leads to a strong edge-11-coloring of $S_n^5$. □

We first establish a useful claim:

Claim 1.

Let $A_i=\{e_i^{\prime },e_i^{\prime \prime }\}\subseteq L\left(x_i\right)$ for $i=0,1,\dots ,n-1$. Let $A=A_0\cup A_1\cup \cdots \cup A_{n-1}$. Then A can be strongly edge-5-colored on the graph $S_n^k$.

Proof. 

Since $n\ge 5$ is odd, we can give an edge 5-coloring $\pi$ of A as follows: coloring $A_1$ with 2, 4; $A_2$ with 3, 5; $A_3$ with 1, 4; each of $A_0,A_5,A_7,\dots ,A_{n-2}$ with 1, 3; and each of $A_4,A_6,A_8,\dots ,A_{n-1}$ with 2, 5. It is easy to confirm that $\pi$ is a strong edge-5-coloring of A restricted in the graph $S_n^k$. □

Lemma 7.

Let $k\ge 6$, and let $n\ge 5$ be odd. Then ${\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)\le \lceil 2.5k-2\rceil$.

Proof. 

If k is even, then by Lemma 6(1) and repeatedly applying Claim 1, we get that ${\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)\le 8+5·\frac{k-4}{2}=2.5k-2=\lceil 2.5k-2\rceil$.

If k is odd, then by Lemma 6(2) and repeatedly applying Claim 1, we get that ${\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)\le 11+5·\frac{k-5}{2}=2.5k-1.5=\lceil 2.5k-2\rceil$. □

3. Outerplanar Graphs

Suppose that G is a connected outerplanar graph. Let $I\subseteq V\left(G\right)$ be a clique-cut of G with $1\le \left|I\right|\le 2$; that is, $G\left[I\right]$ is $K_1$ or $K_2$ such that $G-I$ is disconnected. If $G-I$ has at least two components each containing at least one edge, then I is said to be a separable clique-cut.

For a separable clique-cut I of G, let $H_1,H_2,\dots ,H_{\mathrm{s}}$ ($s\ge 2$) denote the components of $G-I$ with $\left|E\right(H_1\left)\right|\ge 1$ and $\left|E\right(H_2\left)\right|\ge 1$. We set $G_1=G[E\left(H_1\right)\cup E\left(I\right)]$ and $G_2=G[E\left(H_2\right)\cup \cdots \cup E\left(H_{\mathrm{s}}\right)\cup E\left(I\right)]$, where $E\left(I\right)$ denotes the set of edges in G which are incident to at least one vertex in I.

The following lemma plays a crucial role in the proof of our main results.

Lemma 8.

Let G be a connected outerplanar graph with a separable clique-cut I. Suppose that $G_1$ and $G_2$ are defined as above. Then

$${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=max\{{\chi }_{\mathrm{s}}^{\prime }\left(G_1\right),{\chi }_{\mathrm{s}}^{\prime }\left(G_2\right)\}$$

Proof. 

Let $l_1={\chi }_{\mathrm{s}}^{\prime }\left(G_1\right)$, $l_2={\chi }_{\mathrm{s}}^{\prime }\left(G_2\right)$, and $l=max\{l_1,l_2\}$. For $i=1,2$, let ${\varphi }_i$ be a strong edge-l-coloring of $G_i$ using the color set $C=\{1,2,\dots ,l\}$. By the definition of $G_i$, we deduce that $E\left(I\right)\subset E\left(G_i\right)$ for $i=1,2$, and $E\left(G_1\right)\cap E\left(G_2\right)=E\left(I\right)$. Observe that any edge in $E\left(G_1\right)\backslash E\left(I\right)$ and any edge in $E\left(G_2\right)\backslash E\left(I\right)$ have distance at least three in G. Moreover, since the distance between any two edges in $E\left(I\right)$ is less than three, no two edges in $E\left(I\right)$ are assigned same color in both ${\varphi }_1$ and ${\varphi }_2$. So we may assume that ${\varphi }_1\left(e\right)={\varphi }_2\left(e\right)$ for each $e\in E\left(I\right)$. Combining ${\varphi }_1$ and ${\varphi }_2$, we get a strong edge-l-coloring of G. This shows that ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le l$. On the other hand, since $G_i$ is a subgraph of G, we have naturally that ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\ge max\{{\chi }_{\mathrm{s}}^{\prime }\left(G_1\right),{\chi }_{\mathrm{s}}^{\prime }\left(G_2\right)\}=l$. Consequently, ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=l$. □

Theorem 1.

Let G be an outerplanar graph with $\Delta \ge 4$. If G does not contain $F_1$ as a subgraph, then ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 3\Delta -4$.

Proof. 

Assume the contrary, let G be a counterexample with $\left|E\right(G\left)\right|$ being as small as possible. Then G is connected, $\left|E\right(G\left)\right|\ge 3\Delta -3$, and possesses the following properties:

(P1) No $F_1$ is contained in G or its subgraphs.

(P2)G is not strongly edge-$(3\Delta -4)$-colorable, but any subgraph H of G with $\left|E\right(H\left)\right|<\left|E\right(G\left)\right|$ is strongly edge-$(3\Delta -4)$-colorable.

In fact, if $\Delta \left(H\right)<\Delta$, then by Lemma 1, ${\chi }_{\mathrm{s}}^{\prime }\left(H\right)\le 3\Delta \left(H\right)-3\le 3(\Delta -1)-3<3\Delta -4$ since $\Delta \ge 4$. If $\Delta \left(H\right)=\Delta$, then by the minimality of G, ${\chi }_{\mathrm{s}}^{\prime }\left(H\right)\le 3\Delta \left(H\right)-4=3\Delta -4$.

(P3)G is not a tree; otherwise ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 2\Delta -1<3\Delta -4$, contradicting (P2).

By Lemma 8, the following claim holds:

Claim 2.

G does not contain a separable clique-cut $I\subseteq V\left(G\right)$ with $1\le \left|I\right|\le 2$.

Embed G to the plane so that all the vertices lie in the boundary of $f_0\left(G\right)$. Let H denote the graph obtained from G by removing all leaves. By (P3) and Claim 2, we can easily deduce Claims 3 and 4 below.

Claim 3.

H is 2-connected, and $b\left(f_0\left(H\right)\right)$ forms a Hamiltonian cycle. This furthermore implies that all vertices in $V\left(G\right)\backslash V\left(H\right)$ are leaves.

Claim 4.

Every inner edge $uv$ of H is incident to an end-3-face $\left[uvw\right]$ such that $d_G\left(w\right)=d_H\left(w\right)=2$.

Claim 4 implies that $2\le \Delta \left(H\right)\le 4$; for otherwise H will contain an inner edge $xy$ with $d_H\left(x\right)\ge 5$ and $\{x,y\}$ is a separable clique-cut of G.

Let $G^{*}$ denote the graph obtained from G by carrying out repeatedly the following operation:

$(*)$ If x is a 2-vertex of H incident to an end-3-face $\left[xyz\right]$, then we split x into two new vertices $y_1$ and $z_1$ so that $y_1$ joins with y, and $z_1$ joins with z.

Intuitively speaking, every 2-vertex of H which is incident to an end-3-face is replaced by two leaves in G. It is easy to see that $\Delta \left(G^{*}\right)=\Delta \left(G\right)$, and $\varphi$ is a strong edge-k-coloring of $G^{*}$ if and only if $\varphi$ is a strong edge-k-coloring of G.

It is easily observed that $G^{*}$ is a spanning subgraph of some sun-graph $S_n^k$, where $k=\Delta \left(G\right)$ and n is the total number of $3^{+}$-vertices in G and the number of 2-vertices in G which are not on any 3-face. As an example, we observe the graphs G and $G^{*}$ depicted in Figure 2.

Noting that $3k-4\ge max\{2k,\lceil 2.5k-2\rceil \}$, we deduce by Lemmas 5 and 7 that ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)={\chi }_{\mathrm{s}}^{\prime }\left(G^{*}\right)\le {\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)\le 3k-4=3\Delta -4$. This completes the proof of the theorem. □

Combining Theorem 1 and Lemmas 1 and 2(1), the following theorem holds:

Theorem 2.

Let G be an outerplanar graph with $\Delta \ge 4$. Then ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 3\Delta -3$; and ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=3\Delta -3$ if and only if G contains $F_1$ as a subgraph.

Theorem 3.

Let G be an outerplanar graph with maximum degree $\Delta =3$. If G does not contain $F_1,F_2$ or $F_3$ as a subgraph, then ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 5$.

Proof. 

Assume the contrary, let G be a counterexample with $\left|E\right(G\left)\right|$ being as small as possible. Then G is connected, $\left|E\right(G\left)\right|\ge 6$, and possesses the following properties:

(Q1) None of $F_1,F_2,F_3$ is contained in G or its subgraphs.

(Q2)G is not strongly edge-5-colorable, but any subgraph H of G with $\left|E\right(H\left)\right|<\left|E\right(G\left)\right|$ is strongly edge-5- colorable. Actually, if $\Delta \left(H\right)\le 2$, then by Lemma 3, ${\chi }_{\mathrm{s}}^{\prime }\left(H\right)\le 5$. If $\Delta \left(H\right)=3$, then by the minimality of G, we obtain that ${\chi }_{\mathrm{s}}^{\prime }\left(H\right)\le 5$.

(Q3)G is not a tree; otherwise ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 5$, contradicting (Q2).

By Lemma 8, G does not contain a separable clique-cut $I\subseteq V\left(G\right)$ with $1\le \left|I\right|\le 2$.

Embed G to the plane so that all the vertices lie in $b\left(f_0\left(G\right)\right)$. Removing all the leaves of G, we get a subgraph H of G. Similarly to the proof of Theorem 1, we conclude the following:

• H is 2-connected, and all vertices in $V\left(G\right)\backslash V\left(H\right)$ are leaves.
• Every inner edge $uv$ of H is incident to an end-3-face $\left[uvw\right]$ such that $d_G\left(w\right)=d_H\left(w\right)=2$.

Let $G^{*}$ be the graph obtained from G by doing repeatedly the following operation:

$(*)$ If x is a 2-vertex of H incident to an end-3-face $\left[xyz\right]$ in H, then we split x into two new vertices $y_1$ and $z_1$ so that $y_1$ joins with y, and $z_1$ joins with z.

Then $\Delta \left(G^{*}\right)=\Delta \left(G\right)$, and ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)={\chi }_{\mathrm{s}}^{\prime }\left(G^{*}\right)$. Note that $G^{*}$ is a spanning subgraph of some sun-graph $S_n^3$, where n is the total number of 3-vertices in G and the number of 2-vertices in G which are not on any 3-face. By Lemma 4, we derive immediately that ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)={\chi }_{\mathrm{s}}^{\prime }\left(G^{*}\right)\le {\chi }_{\mathrm{s}}^{\prime }\left(S_n^3\right)\le 5$. □

Combining Theorem 3 and Lemmas 1 and 2, we have the following:

Theorem 4.

Let G be an outerplanar graph with $\Delta =3$. Then ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 6$; and ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=6$ if and only if G contains at least one of $F_1,F_2,F_3$ as a subgraph.

When restricted to the family of bipartite outerplanar graphs G, smaller and tight upper bounds for ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)$ can be obtained.

Theorem 5.

Let G be a bipartite and outerplanar graph with maximum degree $\Delta \ge 3$. Then ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 2\Delta$; moreover, ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=2\Delta$ if and only if G contains $F_2$ as a subgraph.

Proof. 

We first show that ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 2\Delta$. Assume the contrary, let G be a counterexample with $\left|E\right(G\left)\right|$ being as small as possible. Then G is connected, other than a tree, and is not strongly edge $2\Delta$-colorable, but any subgraph H of G with $\left|E\right(H\left)\right|<\left|E\right(G\left)\right|$ is strongly edge-$2\Delta$-colorable. Moreover, by Lemma 8, there is no separable clique-cut $I\subseteq V\left(G\right)$ with $1\le \left|I\right|\le 2$.

Embed G to the plane so that all the vertices occur in $b\left(f_0\left(G\right)\right)$. Removing all the leaves of G, we obtain a subgraph H of G. Then H is a Hamiltonian cycle without chords, and $V\left(G\right)\backslash V\left(H\right)$ are all leaves. So G is a subgraph of some $S_n^k$ where $n=\left|V\right(H\left)\right|$ is even and $k=\Delta \left(G\right)$. By Lemma 5, ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le {\chi }_{\mathrm{s}}^{\prime }\left(S_n^k\right)\le 2k=2\Delta$.

If G contains $F_2$ as a subgraph, then ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\ge {\chi }_{\mathrm{s}}^{\prime }\left(F_2\right)=\left|E\left(F_2\right)\right|=2\Delta$. Using the foregoing proof, we get that ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)=2\Delta$. Conversely, if G does not contain $F_2$ as a subgraph, then similarly to the above discussion we can show that ${\chi }_{\mathrm{s}}^{\prime }\left(G\right)\le 2\Delta -1$. □