Injective chromatic number of outerplanar graphs

TL;DR

This paper investigates the injective chromatic number of outerplanar graphs, establishing tight bounds based on maximum degree and girth, and providing exact values for specific subclasses.

Contribution

It derives new tight bounds for the injective chromatic number of outerplanar graphs considering maximum degree and girth, including exact values for certain cases.

Findings

For all outerplanar graphs, hi_i(G) st; bound is tight.

For elta=3, hi_i(G) st+1; tight for girth 3 and 4.

For 2-connected outerplanar graphs with elta=3, g , and elta , g , hi_i(G) = elta.

Abstract

An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by $\chi_i(G)$. In this paper, the injective chromatic number of outerplanar graphs with maximum degree $\Delta$ and girth $g$ is studied. It is shown that for every outerplanar graph, $\chi_i(G)\leq \Delta+2$, and this bound is tight. Then, it is proved that for outerplanar graphs with $\Delta=3$, $\chi_i(G)\leq \Delta+1$ and the bound is tight for outerplanar graphs of girth three and $4$. Finally, it is proved that, the injective chromatic number of $2-$connected outerplanar graphs with $\Delta=3$, $g\geq 6$ and $\Delta\geq 4$, $g\geq 4$ is equal to $\Delta$.

Full text

Keywords: Injective coloring, Injective chromatic number, Outerplanar graph.

1 Introduction

All graphs we have considered here are finite, connected and simple. A plane graph is a planar drawing of a planar graph in the Euclidean plane. The vertex set, edge set, face set, minimum degree and maximum degree of a plane graph G, are denoted by $V(G)$, $E(G)$, $F(G)$, $\delta(G)$ and $\Delta(G)$, respectively. A vertex of degree $k$ is called a $k-$vertex. For vertex $v\in V(G)$, $N_{G}(v)$ is the set of neighbors of $v$ in $G$. The girth of a graph G, $g(G)$, is the length of a shortest cycle in $G$. If there is no confusion, we delete $G$ in the notations. A face $f\in F(G)$ is denoted by its boundary walk $f=[v_{1}v_{2}\ldots v_{k}]$, where $v_{1},v_{2},\ldots,v_{k}$ are its vertices in the clockwise order. Also, the vertices $v_{1}$ and $v_{k}$ as end vertices of $f$ are denoted by ${v_{{}_{L}}}_{{}_{{}_{f}}}$ and ${v_{{}_{R}}}_{{}_{{}_{f}}}$, respectively. An outerplanar graph is a graph with a planar drawing for which all vertices belong to the outer face of the drawing. It is known that a graph $G$ is an outerplanar graph if and only if $G$ has no subdivision of complete graph $K_{4}$ and complete bipartite graph $K_{2,3}$. A path $P:v_{1},v_{2},\ldots,v_{k}$ is called a simple path in $G$ if $v_{2},\ldots,v_{k-1}$ are all $2-$vertices in $G$. The length of a path is the number of its edges. We say that a face $f=[v_{1}v_{2}\ldots v_{k}]$ is an end face of an outerplane graph $G$, if $P:v_{1},v_{2},\ldots,v_{k}$ is a simple path in $G$. An end block in graph $G$ is a maximal $2-$connected subgraph of $G$ contains an unique cut vertex of $G$.

A proper $k-$coloring of a graph $G$ is a mapping from $V(G)$ to the set of colors $\{1,2,\ldots,k\}$ such that any two adjacent vertices have different colors. The chromatic number, $\chi(G)$, is a minimum integer $k$ that $G$ has a proper $k-$coloring. A coloring $c$ of $G$ is called an injective coloring if for every two vertices $u$ and $v$ which have common neighbor, $c(u)\neq c(v)$. That means, the restriction of $c$ to the neighborhood of any vertex is an injective function. The injective chromatic number, $\chi_{i}(G)$, is the least integer $k$ such that $G$ has an injective $k-$coloring. Note that an injective coloring is not necessarily a proper coloring. In fact, $\chi_{i}(G)=\chi(G^{(2)})$, where $V(G^{(2)})=V(G)$ and $uv\in E(G^{(2)})$ if and only if $u$ and $v$ have a common neighbor in $G$. The square of graph $G$, denoted by $G^{2}$, is a graph with vertex set $V(G)$, where two vertices are adjacent in $G^{2}$ if and only if they are at distance at most two in $G$. Since $G^{(2)}$ is a subgraph of $G^{2}$, obviously, $\chi_{i}(G)\leq\chi(G^{2})$. The concept of injective coloring is introduced by Hahn et al. in $2002$ [7]. It is clear that for every graph $G$, $\chi_{i}(G)\geq\Delta$. In general, in [7] Hahn et al. proved that $\Delta\leq\chi_{i}(G)\leq\Delta^{2}-\Delta+1$.

In [13], Wegner expressed the below conjecture for the chromatic number of the square of planar graphs.

Conjecture 1

.* [13] If $G$ is a planar graph with maximum degree $\Delta$, then*

• •

For $\Delta=3$, $\chi(G^{2})\leq\Delta+2$.

• •

For $4\leq\Delta\leq 7$, $\chi(G^{2})\leq\Delta+5$.

• •

For $\Delta\geq 8$, $\chi(G^{2})\leq\lfloor\frac{3\Delta}{2}\rfloor+1$.

Since $\chi_{i}(G)\leq\chi(G^{2})$, Lužar and Škrekovski in [10] proposed the following conjecture for the injective chromatic number of planar graphs.

Conjecture 2

.* [10] If $G$ is a planar graph with maximum degree $\Delta$, then*

• •

For $\Delta=3$, $\chi_{i}(G)\leq\Delta+2$.

• •

For $4\leq\Delta\leq 7$, $\chi_{i}(G)\leq\Delta+5$.

• •

For $\Delta\geq 8$, $\chi_{i}(G)\leq\lfloor\frac{3\Delta}{2}\rfloor+1$.

The injective coloring of planar graphs with respect to its girth and maximum degree is studied in [1, 2, 3, 4, 5, 6, 9, 11]. In [8], Lih and Wang proved upper bound $\Delta+2$ for the chromatic number of square of outerplanar graphs.

Theorem 1

.* [8] If $G$ is an outerplanar graph, then $\chi(G^{2})\leq\Delta+2$.*

Since $\chi_{i}(G)\leq\chi(G^{2})$, Conjecture 2 is true for outerplanar graphs.

Corollary 1

.* If $G$ is an outerplanar graph, then $\chi_{i}(G)\leq\Delta+2$.*

In Figure 1, an outerplanar graph with $\Delta=4$, $g=3$ and $\chi_{i}(G)=\Delta+2=6$ is shown. Therefore, the given bound in Corollary 1 is tight.

In this paper, we study the injective chromatic number of outerplanar graphs. The main results of Section $2$ are as follows.

If $G$ is an outerplanar graph with maximum degree $\Delta$ and girth $g$, then

• •

For $\Delta=3$ and $g\geq 3$, $\chi_{i}(G)\leq\Delta+1=4$. (Theorem 2)

• •

For $\Delta=3$ and $g\geq 5$, with no face of degree $k$, $k\equiv 2$ (mod $4$), $\chi_{i}(G)=\Delta$. (Theorem 3)

• •

For $\Delta=3$ and $g\geq 6$, $\chi_{i}(G)=\Delta$. (Theorem 5)

• •

For $\Delta\geq 4$ and $g\geq 4$, $\chi_{i}(G)=\Delta$. (Theorems 6 and 8)

2 Main Results

First, we prove a tight bound for the injective chromatic number of outerplanar graphs with $\Delta=3$. Note that if $\Delta=2$, then $G$ is an union of paths and cycles, which obviously $\chi_{i}(G)\leq 3=\Delta+1$. Moreover, if $G$ is a path or an even cycle, then $\chi_{i}(G)=2$; and if $G$ is an odd cycle, then $\chi_{i}(G)=3$ [7].

Theorem 2

.* If $G$ is an outerplanar graph with $\Delta=3$ and $g\geq 3$, then $G$ has a $4-$injective coloring such that in every simple path of lenght three, at most three colors appear. Moreover, the bound is tight.*

**Proof. **

We prove the theorem by the induction on $|V(G)|$. In Figure 2, all outerplanar graphs with $\Delta=3$ and $g\geq 3$ of order $4$ and $5$ with an injective coloring with desired property are shown. Obviously, in the left side graph, $\chi_{i}(G)=4$. Hence, bound $\Delta+1$ is tight.

Now suppose that $G$ is an outerplane graph with $\Delta=3$, $g\geq 3$ and the statement is true for all outerplanar graphs with $\Delta=3$ and $g\geq 3$ of order less than $|V(G)|$. The following two cases can be caused.

If an end block of $G$ is an edge, say $uv$, where $\deg(u)=1$, then we consider the maximal simple path $P:(v_{1}=u),(v_{2}=v),v_{3},\ldots,v_{k}$ in $G$. Since $P$ is a maximal simple path and $\Delta(G)=3$, we have $\deg(v_{k})=3$. Suppose that $N(v_{k})=\{w_{1},w_{2},v_{k-1}\}$ and $c$ is a $4-$injective coloring of $G\setminus\{v_{1},v_{2},\ldots,v_{k-1}\}$ with colors $\{\alpha,\beta,\gamma,\lambda\}$ such that every simple path of length three has at most three colors. Note that $w_{1}$ and $w_{2}$ have a common neighbor $v_{k}$ therefore, $c(w_{1})\neq c(w_{2})$. In this case, we assign to the ordered vertices $v_{k-1},v_{k-2},\ldots,v_{2},v_{1}$ of path $P$ the ordered string $(ssttsstt\ldots)$ where, $s\in\{\alpha,\beta,\gamma,\lambda\}\setminus\{c(v_{k}),c(w_{1}),c(w_{2})\}$ and $t=c(v_{k})$.

If the minimum degree of every end block of $G$ is at least two in $G$, then we consider an end face $f=[v_{i}v_{i+1}\ldots v_{j}]$ in an end block $B$ of $G$ in clockwise order, where $v_{1}$ is the vertex cut of $G$ belongs to $B$. Let $H$ be the induced subgraph of $G$ on $2-$vertices of $f$. By the induction hypothesis $G\setminus H$ has a $4-$injective coloring $c$ with colors $\{\alpha,\beta,\gamma,\lambda\}$, such that every simple path of length three has at most three colors. Hence, in $G\setminus H$ at most three colors are used for vertices $v_{i-1},v_{i},v_{j},v_{j+1}$. Now we extend $c$ to an injective coloring of $G$ with the desired property.

If $c(v_{i})=c(v_{j})$, then we assign to the ordered vertices $v_{i+1},v_{i+2},\ldots,v_{j-1}$ the ordered string $(ssttsstt\ldots)$ where, $s\in\{\alpha,\beta,\gamma,\lambda\}\setminus\{c(v_{i-1}),c(v_{i})=c(v_{j}),c(v_{j+1})\}$ and $t\in\{\alpha,\beta,\gamma,\lambda\}\setminus\{c(v_{i})=c(v_{j}),c(v_{j+1}),s\}$. If $v_{i}=v_{j}$ and $j-i-1\equiv 1,2$ (mod $4$), then change the color of $v_{j-1}$ to $t^{\prime}\in\{\alpha,\beta,\gamma,\lambda\}\setminus\{c(v_{i-1})=c(v_{j+1}),s,t\}$.

If $c(v_{i})\neq c(v_{j})$, then we assign to the ordered vertices $v_{i+1},v_{i+2},\ldots,v_{j-1}$ the ordered string $(ssttsstt\ldots)$ where, $s\in\{\alpha,\beta,\gamma,\lambda\}\setminus\{c(v_{i-1}),c(v_{i}),c(v_{j}),c(v_{j+1})\}$. If $j-i-1\equiv 1,2$ (mod $4$), then $t\in\{\alpha,\beta,\gamma,\lambda\}\setminus\{c(v_{j}),s\}$. If $j-i-1\equiv 0,3$ (mod $4$), then $t\in\{\alpha,\beta,\gamma,\lambda\}\setminus\{c(v_{i}),c(v_{j+1}),s\}$. In the case $j-i-1\equiv 0$ (mod $4$), if $t=c(v_{j})$, then change the color $v_{j-2}$ to $t^{\prime}\in\{\alpha,\beta,\gamma,\lambda\}\setminus\{c(v_{j})=t,s\}$. Note that, since by the induction hypothesis $|\{c(v_{i-1}),c(v_{i}),c(v_{j}),c(v_{j+1})\}|\leq 3$, in each cases the colors $s$ and $t$ exist. It can be easily seen that the given coloring is a $4-$injective coloring for $G$ such that every simple path of length three in $G$ has at most three colors as well. *  *

Graph $G$ in Figure 3 is an outerplanar graph of girth $4$ with maximum degree three and injective chromatic number $4$. Since each pair of set $\{u,v,w\}$ have a common neighbor, in every injective coloring of $G$, they must have three different colors. In the similar way, we need three different colors for the vetrices $\{x,y,z\}$. Without loss of generality, color the vertices $u,v,w$ with color $\alpha$, $\beta$ and $\gamma$, respectively. Now by devoting any permutation of these colors to vertices $x,y$ and $z$, it can be checked that in each case we need a new color for the other vertices. Therefore, bound $\Delta+1$ in Theorem 2 is tight for outerplanar graphs with $\Delta=3$, $g=4$ and $g=3$ (see also Figure 2).

In the next theorems, we improve bound $\Delta+1$ to $\Delta$ for outerplanar graphs with $\Delta=3$ of girth greater than $4$.

Theorem 3

.* If $G$ is a $2-$connected outerplanar graph with $\Delta=3$, $g\geq 5$ and no face of degree $k$, where $k\equiv 2$ (mod $4$), then $G$ has a $3-$injective coloring such that in every simple path of lenght three, exactly three colors appear.*

**Proof. **

Since $\chi_{i}(G)\geq\Delta(G)$, it is enough to show that $\chi_{i}(G)\leq\Delta(G)$. We prove it by the induction on $|V(G)|$. In Figure 4, the $2-$connected outerplanar graph with $\Delta=3$ and $g\geq 5$ of order $8$ with an injective coloring with desired property is shown.

Now suppose that $G$ is a $2-$connected outerplane graph with $\Delta=3$, $g\geq 5$ and no face of degree $k$, where $k\equiv 2$ (mod $4$) and the statement is true for all such $2-$connected outerplanar graphs of order less than $|V(G)|$.

Let $f=[v_{i}v_{i+1}\ldots v_{j}]$ be an end face of $G$ in clockwise order and $H$ be the induced subgraph of $G$ on $2-$vertices of $f$. If $\Delta(G\setminus H)=3$, then by the induction hypothesis $G\setminus H$ has a $3-$injective coloring $c$ with colors $\{\alpha,\beta,\gamma\}$, such that every simple path of length three has exactly three colors.

If $\Delta(G\setminus H)=2$, then we color the vertices of $G\setminus H$ as follows. If $G\setminus H=C_{k}$, where $k>5$ and $k\equiv 0,1$ (mod $3$), then color the ordered vertices $v_{i-1},v_{i},v_{j},v_{j+1},\ldots,v_{i-2}$ with the ordered string $(\alpha\beta\gamma\alpha\beta\gamma\ldots)$. If $k>5$ and $k\equiv 2$ (mod $3$), then color the ordered vertices $v_{i-1},v_{i},v_{j},v_{j+1},\ldots,v_{i-5}$ with the ordered string $(\alpha\beta\gamma\alpha\beta\gamma\ldots)$. Then color the vertices $v_{i-4}$, $v_{i-3}$ and $v_{i-2}$ with colors $\beta$, $\gamma$ and $\alpha$, respectively. One can check that every simple path of length three in $G\setminus H$ has exactly three colors. If $G\setminus H=C_{5}$, then since $|V(G)|>8$, $f=[v_{i}v_{i+1}\ldots v_{j}]$ is a cycle of length at least $6$. In this case, we consider the end face $f^{\prime}=[v_{j}v_{j+1}\ldots v_{i}]$ and follow the above proof when $H$ is induced subgraph of $G$ on $2-$vertices of $f^{\prime}$. In the following, we extend injective coloring $c$ of $G\setminus H$ to an injective coloring of $G$ with the desired property.

If $c(v_{i})=c(v_{j})$, then we assign to the ordered vertices $v_{i+1},v_{i+2},\ldots,v_{j-1}$ the ordered string $(s_{1}s_{2}s_{3}s_{4}s_{1}s_{2}s_{3}s_{4}\ldots)$, where $s_{1}=c(v_{j+1})$. Since, $G$ has no face of degree $k$ where $k\equiv 2$ (mod $4$), we have following cases. If $j-i-1\equiv 1$ (mod $4$), then let $s_{2}=c(v_{i-1})$, $s_{3}=s_{4}=c(v_{i})=c(v_{j})$ and change the color of vertices $v_{j-2}$ and $v_{j-1}$ to $c(v_{j+1})$ and $c(v_{i-1})$, respectively. If $j-i-1\equiv 2$ (mod $4$), then let $s_{2}=s_{1}$, $s_{3}=c(v_{i-1})$ and $s_{4}=c(v_{i})=c(v_{j})$ and change the color of $v_{j-1}$ to $c(v_{i-1})$. If $j-i-1\equiv 3$ (mod $4$), then let $s_{2}=s_{1}$, $s_{3}=c(v_{i-1})$ and $s_{4}=c(v_{i})=c(v_{j})$.

If $c(v_{i})\neq c(v_{j})$ and $c(v_{i-1})=c(v_{j+1})$, then we assign to the ordered vertices $v_{i+1},v_{i+2},\linebreak{\ldots,v_{j-1}}$ the ordered string $(s_{1}s_{2}s_{3}s_{4}s_{1}s_{2}s_{3}s_{4}\ldots)$, where $s_{1}=c(v_{i})$. If $j-i-1\equiv 1,2$ (mod $4$), then $s_{2}=c(v_{j})$ and $s_{3}=s_{4}=c(v_{i-1})=c(v_{j+1})$. In the case $j-i-1\equiv 1$ (mod $4$), we change the color of vertices $v_{j-2}$ and $v_{j-1}$ to $c(v_{i})$ and $c(v_{j})$, respectively. If $j-i-1\equiv 3$ (mod $4$), then let $s_{2}=c(v_{i-1})=c(v_{j+1})$ and $s_{3}=s_{4}=c(v_{j})$.

If $c(v_{i})\neq c(v_{j})$ and $c(v_{i-1})=c(v_{i})$, then we assign to the ordered vertices $v_{i+1},v_{i+2},\ldots,\linebreak{v_{j-1}}$ the ordered string $(s_{1}s_{2}s_{3}s_{4}s_{1}s_{2}s_{3}s_{4}\ldots)$, where $s_{1}=c(v_{j+1})$. If $j-i-1\equiv 1$ (mod $4$), then let $s_{2}=c(v_{j})$, $s_{3}=c(v_{i})=c(v_{i-1})$, $s_{4}=s_{1}$ and change the color of vertex $v_{j-1}$ to $c(v_{j})$. If $j-i-1\equiv 2$ (mod $4$), then let $s_{2}=c(v_{j})$ and $s_{3}=s_{4}=c(v_{i-1})=c(v_{i})$. If $j-i-1\equiv 3$ (mod $4$), then we assign to the ordered vertices $v_{j-1},v_{j-2},\ldots,v_{i+1}$ the ordered string $(s_{1}s_{2}s_{3}s_{4}s_{1}s_{2}s_{3}s_{4}\ldots)$, where $s_{1}=c(v_{j})$, $s_{2}=c(v_{j+1})$, $s_{3}=s_{4}=c(v_{i})=c(v_{i-1})$ and change the colors of $v_{i+1}$ to $c(v_{j+1})$.

If $c(v_{i})\neq c(v_{j})$ and $c(v_{j})=c(v_{j+1})$, then we assign to the ordered vertices $v_{j-1},v_{j-2},\ldots,\linebreak{v_{i+1}}$ the ordered string $(s_{1}s_{2}s_{3}s_{4}s_{1}s_{2}s_{3}s_{4}\ldots)$, where $s_{1}=c(v_{i-1})$. If $j-i-1\equiv 1$ (mod $4$), then $s_{2}=c(v_{i})$, $s_{3}=c(v_{j+1})=c(v_{j})$, $s_{4}=s_{1}$ and change the color of $v_{i+1}$ to $c(v_{i})$. If $j-i-1\equiv 2$ (mod $4$), then $s_{2}=c(v_{i})$ and $s_{3}=s_{4}=c(v_{j+1})$. If $j-i-1\equiv 3$ (mod $4$), then we assign to the ordered vertices $v_{i+1},v_{i+2},\ldots,v_{j-1}$ the ordered string $(s_{1}s_{2}s_{3}s_{4}s_{1}s_{2}s_{3}s_{4}\ldots)$ where, $s_{1}=c(v_{i})$, $s_{2}=c(v_{i-1})$ and $s_{3}=s_{4}=c(v_{j})=c(v_{j+1})$ and change the color of $v_{j-1}$ to $c(v_{i-1})$. It can be seen that the given coloring is a $4-$injective coloring for $G$ such that every simple path of length three in $G$ has at most three colors.*  *

In Theorem 5, we improve bound $\Delta+1$ in Theorem 2 to $\Delta$ for outerplanar graph with $\Delta=3$ and $g\geq 6$. First, we need the following theorem.

Theorem 4

.* [12] Let $G$ be a connected graph and $L$ be a list-assignment to the vertices, where $|L(v)|\geq\deg(v)$ for each $v\in V(G)$. If*

$|L(v)|>\deg(v)$* for some vertex $v$, or* 2. 2.

$G$* contains a block which is neither a complete graph nor an induced odd cycle,*

then $G$ admits a proper coloring such that the color assign to each vertex $v$ is in $L(v)$.

Theorem 5

.* If $G$ is an outerplanar graph with $\Delta=3$ and $g\geq 6$, then $\chi_{i}(G)=\Delta$.*

**Proof. **

Since $\chi_{i}(G)\geq\Delta$, it is enough to show that $\chi_{i}(G)\leq\Delta$. Let $G$ be a minimal counterexample for this statement. That means $G$ is an outerplane graph with $\Delta=3$, $g\geq 6$ and $\chi_{i}(G)\geq\Delta+1$, such that every proper subgraph of $G$ has a $\Delta-$injective coloring. Obviously $\delta(G)\geq 2$. Now consider an end face $f=[v_{i}v_{i+1}\ldots v_{j}]$ in an end block $B$ of $G$ in clockwise order, where $v_{1}$ is the vertex cut of $G$ belongs to $B$. Since $\Delta=3$ and $g\geq 6$, the degree of face $f$ is at least $6$ and the degree of $v_{i}$ and $v_{j}$ are three. If $\Delta(G\setminus H)=3$, then by the minimality of $G$, we have $\chi_{i}(G\setminus H)\leq\Delta(G\setminus H)\leq\Delta(G)$. Also, if $G\setminus H$ is a cycle, then $\chi_{i}(G\setminus H)\leq 3=\Delta$.

Now, we extend the $\Delta-$injective coloring of $G\setminus H$ to a $\Delta-$injective coloring of $G$, which contradict our assumption. Each of the vertices $v_{i}$ and $v_{j}$ has at most $\Delta-1=2$ neighbors except $v_{i+1}$ and $v_{j-1}$, respectively. Hence, for each of vertices $v_{i+1}$ and $v_{j-1}$ there is at least one available color. Also, among the colored vertices in $G\setminus H$, the only forbbiden colors for vertices $v_{i+2}$ and $v_{j-2}$ are colors of the vertices $v_{i}$ and $v_{j}$, respectively. The other vertices have three available colors. Now consider induced subgraph of $G^{(2)}$ on the vertices of $H$, denoted by $G^{(2)}[H]$, and list of available colors for each vertex of $H$. The components of $G^{(2)}[H]$ are some paths or cycles satisfying the assumption of Theorem 4. Thus, we have a proper $\Delta-$coloring for $G^{(2)}[H]$ using the available colors which is a $\Delta-$injective coloing of $H$ as desired.*  *

Now we are ready to determine the injective chromatic number of $2-$connected outerplanar graphs with maximum degree and girth greater than three. We prove this fact by two different methods for the cases $\Delta=4$ and $\Delta\geq 5$.

Theorem 6

.* If $G$ is a $2-$connected outerplanar graph with $\Delta=4$ and $g\geq 4$, then $G$ has a $4-$injective coloring $c$ such that for every adjacent vertices $v$ and $u$ of degree three with $N(v)=\{u,v_{1},v_{2}\}$ and $N(u)=\{v,u_{1},u_{2}\}$, $\{c(u),c(v_{1}),c(v_{2})\}\neq\{c(v),c(u_{1}),c(u_{2})\}$.*

**Proof. **

Since $\chi_{i}(G)\geq\Delta(G)=4$, it is enough to show that $\chi_{i}(G)\leq 4$. We prove it by the induction on $|V(G)|$. In Figure 5, the $2-$connected outerplanar graph with $\Delta=4$ and $g\geq 4$ of order $8$ with an injective coloring of desired property is shown.

Now suppose that $G$ is a $2-$connected outerplane graph with $\Delta=4$, $g\geq 4$ and the statement is true for all $2-$connected outerplanar graphs with $\Delta=4$ and $g\geq 4$ of order less than $|V(G)|$.

Let $f=[v_{i}v_{i+1}\ldots v_{j}]$ be an end face of $G$ in clockwise order. If $\deg(v_{i})=\deg(v_{j})=3$, then consider induced subgraph $H$ on $2-$vertices of face $f$. Thus, $G\setminus H$ is a $2-$connected outerplane graph with $\Delta(G\setminus H)=4$ and $g(G\setminus H)\geq 4$. Hence, by the induction hypothesis, $G\setminus H$ has a $4-$injective coloring such that for every adjacent vertices $v$ and $u$ of degree three with $N(v)=\{u,v_{1},v_{2}\}$ and $N(u)=\{v,u_{1},u_{2}\}$, $\{c(u),c(v_{1}),c(v_{2})\}\neq\{c(v),c(u_{1}),c(u_{2})\}$. If there are exactly four colors in $\{c(v_{i-1}),c(v_{i}),c(v_{j}),c(v_{j+1})\}$, then consider graph $G^{(2)}[H]$ and list of available colors for each vertex of $H$. Graph $G^{(2)}[H]$ satisfy the assumption of Theorem 4. Thus, we have a $\Delta-$coloring for $G^{(2)}[H]$ which is a $\Delta-$injective coloring of $G$. If there are at most three colors in $\{c(v_{i-1}),c(v_{i}),c(v_{j}),c(v_{j+1})\}$, then color $v_{i+1}$ with one of its colors not in $\{c(v_{i-1}),c(v_{i}),c(v_{j}),c(v_{j+1})\}$ and color $v_{j-1}$ with one of its available colors such that $c(v_{i+1})\neq c(v_{j-1})$. Then color the other vertices of $H$ with one of their available colors similar to above. It can be easily seen that for every adjacent vertices $v$ and $u$ of degree three with $N(v)=\{u,v_{1},v_{2}\}$ and $N(u)=\{v,u_{1},u_{2}\}$, $\{c(u),c(v_{1}),c(v_{2})\}\neq\{c(v),c(u_{1}),c(u_{2})\}$.

Now suppose that each face of $G$ has an end vertex of degree $4$. We have two following cases.

$(i)$ There is an end face $f$ with one end vertex of degree $4$ and the other one of degree less than $4$.

$(ii)$ For each end face $f$, two its end vertices are of degree $4$.

In the former case, suppose that $G$ has an end face $f=[v_{i}v_{i+1}\ldots v_{j}]$, where $\deg(v_{i})=4$ and $\deg(v_{j})=3$. Consider induced subgraph $H$ on $2-$vertices of face $f$. If $\Delta(G\setminus H)=4$, then by the induction hypothesis, $G\setminus H$ has a $4-$injective coloring such that for every adjacent vertices $v$ and $u$ of degree three with $N(v)=\{u,v_{1},v_{2}\}$ and $N(u)=\{v,u_{1},u_{2}\}$, $\{c(u),c(v_{1}),c(v_{2})\}\neq\{c(v),c(u_{1}),c(u_{2})\}$. Now we extend the $4-$injective coloring of $G\setminus H$ to $G$. If $\deg(v_{j+1})=3$, then suppose that $v_{s}$ is the other neighbor of $v_{j+1}$ except $v_{j}$ and $v_{j+2}$. If there are exactly three colors in $\{c(v_{i}),c(v_{j}),c(v_{j+1}),c(v_{j+2}),c(v_{s})\}$, then color vertex $v_{j-1}$ with one of its colors not in $\{c(v_{i}),c(v_{j}),c(v_{j+1}),c(v_{j+2}),c(v_{s})\}$ and color the other vertices of $H$ with one of their available colors as explained in above. If $|\{c(v_{i}),c(v_{j}),c(v_{j+1}),c(v_{j+2}),c(v_{s})\}|=4$ or $\deg(v_{j+1})\neq 3$, then by Theorem 4 color the vertices of $H$ with one of their available colors such that obtained coloring is a $4-$injective coloring of $G$.

If $\Delta(G\setminus H)=3$, then by assumption $(i)$, there is another unique end face, say $f^{\prime}$, with a common neighbor with $f$. Consider induced subgraph $H$ on $2-$vertices of face $f$ and $f^{\prime}$. Thus, $G\setminus H$ is a cycle and $\chi_{i}(G\setminus H)\leq 3$. Now each vertices of $H$ has at least two available colors. Hence, by applying Theorem 4, we obtain a $4-$injective coloring of $G$. Note that, since $g(G)\geq 4$, in this case there is no two adjacent vertices of degree three.

In the latter case, consider the induced subgraph $H$ on $2-$vertices of $f=[v_{i}v_{i+1}\ldots v_{j}]$, where $\deg(v_{i})=\deg(v_{j})=4$. Since $\deg(v_{i})=4$, $G\setminus H$ has an end face $f^{\prime}$ with two ends of degree $4$. Hence, $\Delta(G\setminus H)=4$ and by the induction hypothesis, $G\setminus H$ has a $4-$injective coloring such that for every adjacent vertices $v$ and $u$ of degree three with $N(v)=\{u,v_{1},v_{2}\}$ and $N(u)=\{v,u_{1},u_{2}\}$, $\{c(u),c(v_{1}),c(v_{2})\}\neq\{c(v),c(u_{1}),c(u_{2})\}$. Now by Theorem 4, color the vertices of $H$ with their available colors such that obtained coloring is a $4-$injective coloring of $G$. Obviously, for every adjacent vertices $v$ and $u$ of degree three with $N(v)=\{u,v_{1},v_{2}\}$ and $N(u)=\{v,u_{1},u_{2}\}$, $\{c(u),c(v_{1}),c(v_{2})\}\neq\{c(v),c(u_{1}),c(u_{2})\}$.

•  *

Now we consider $2-$connected outerplanar graphs with $\Delta=5$ and $g\geq 4$. First, we need to prove the following theorem on the structure of $2-$connected outerplanar graphs.

Theorem 7

.* If $G$ is a $2-$connected outerplanar graph, then $G$ has an end face $f=[v_{i}v_{i+1}\ldots v_{j}]$, where either $\deg(v_{i})<5$ or $\deg(v_{j})<5$.*

**Proof. **

First replace every simple path in boundary of each end face of $G$ with a path of length two and name this graph $G^{\prime}$. Graph $G^{\prime}$ is also a $2-$connected outerplane graph that each end face of $G^{\prime}$ is of degree three. Now, let $C:v_{1}v_{2}\ldots v_{n}$ be a Hamilton cycle of $G^{\prime}$ in clockwise order and $f=[v_{i}v_{i+1}v_{i+2}]$ be an end face of $G^{\prime}$. If $G^{\prime}$ is a cycle, then we are done. Hence, suppose that $\Delta(G^{\prime})\geq 3$ and $f=[v_{i}v_{i+1}v_{i+2}]$ is an end face of $G^{\prime}$ where, $\deg(v_{i})$ and $\deg(v_{i+2})$ are at least 5. In what follows, we present an algorithm that find an end face of $G^{\prime}$ such that the degree of at least one of its end vertices is less than $5$. Since by assumption $\deg(v_{i+2})\geq 5$, $v_{i+2}$ has at least two other neighbors except $v_{i}$, $v_{i+1}$ and $v_{i+3}$, named $v_{i^{\prime}}$ and $v_{j^{\prime}}$ where, $j^{\prime}<i^{\prime}$.

Note that the indices of neighbors of all vertices $v_{k}$, $i+3\leq k<i^{\prime}$, is less than $i^{\prime}$; otherwise there is a subdivision of $K_{4}$ on $G^{\prime}$ and it is a contradiction with the assumption that $G^{\prime}$ is an outerplanar graph. Since in each step of the algorithm the indices of vertices in $f_{k}$ are increasing, the algorithm terminates and its output, say $f=[v_{s}v_{s+1}v_{s+2}]$, is an end face in $G^{\prime}$ where $v_{s+2}$ has degree at most $4$. Therefore, by returning the contracted paths to $G^{\prime}$; we have an end face of $G$ that one of its ends is of degree less than $5$.*  *

Theorem 8

.* If $G$ is a $2-$connected outerplanar graph with $\Delta\geq 5$ and $g\geq 4$, then $\chi_{i}(G)=\Delta$.*

**Proof. **

Since $\chi_{i}(G)\geq\Delta(G)$, it is enough to show that $\chi_{i}(G)\leq\Delta(G)$. We prove it by the induction on $|V(G)|$. In Figure 6, the $2-$connected outerplanar graph with $\Delta\geq 5$ and $g\geq 4$ of order $10$ with a $\Delta-$injective coloring is shown.

Now suppose that $G$ is a $2-$connected outerplane graph with $\Delta\geq 5$, $g\geq 4$ and the statement is true for all $2-$connected outerplanar graphs with $\Delta\geq 5$ and $g\geq 4$ of order less that $|V(G)|$. By Theorem 7, $G$ has an end face $f$ of degree at least $4$ such that at least one of its end vertices is of degree at most $4$. Now consider the induced subgraph $H$ on $2-$vertices of end face $f$. If $\Delta(G\setminus H)\geq 5$, then by induction hypothesis, $\chi_{i}(G\setminus H)=\Delta(G\setminus H)\leq\Delta(G)$. If $\Delta(G\setminus H)=4$, then by Theorem 6, $G\setminus H$ has a $4-$injective coloring. Now consider the end face $f=[v_{i}v_{i+1}\ldots v_{j}]$ and suppose that $\deg(v_{i})\leq\Delta$ and $\deg(v_{j})\leq 4$. Since $\Delta\geq 5$, the vertices $v_{i+1}$ and $v_{j-1}$ have at least one and two available colors, respectively. The other vertices of $H$ has at least three available colors. Now consider the graph $G^{(2)}[H]$ and list of available colors for each vertex of H. It can be easily seen that $G^{(2)}[H]$ is union of paths and isolated vertices, which satisfy the assumption of Theorem 4. Hence, $G^{(2)}[H]$ can be colored by at most $\Delta$ colors and the obtained coloring is a $\Delta-$injective coloring of $H$.*  *

**Remark **

.* We believe that, laboriously, the results of Theorems 3, 6 and 8 can be generalized for the outerplanar graphs containing some cut vertices.*