A tight upper bound on the (2,1)-total labeling number of outerplanar graphs

In the channel/frequency assignment problems, we need to assign different frequencies to ‘close’ transmitters so that they can avoid interference. The $L(p,q)$-labelings of a graph have been widely studied as one of important graph theoretical models of this problem. An $L(p,q)$-labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ to the set $\{0,1,\ldots,k\}$ of nonnegative integers such that $|f(x)-f(y)|\geq p$ if $x$ and $y$ are adjacent and $|f(x)-f(y)|\geq q$ if $x$ and $y$ are at distance 2, for all $x$ and $y$ in $V(G)$. We call this invariant (i.e., the minimum value $k$) the $L(p,q)$-labeling number and denote it by $\lambda_{p,q}(G)$. Notice that we can use $k+1$ different labels when $\lambda_{p,q}(G)=k$ since we can use 0 as a label for conventional reasons. We can find many results on $L(p,q)$-labelings in comprehensive surveys by Calamoneri [3] and by Yeh [16].

In [15], Whittlesey et al. studied the $L(2,1)$-labeling number of incidence graphs, where the incidence graph of a graph $G$ is the graph obtained from $G$ by replacing each edge $(v_{i},v_{j})$ with two edges $(v_{i},v_{ij})$ and $(v_{ij},v_{j})$ introducing one new vertex $v_{ij}$. Observe that an $L(p,1)$-labeling of the incidence graph of a given graph $G$ can be regarded as an assignment $f$ from $V(G)\cup E(G)$ to the set $\{0,1,\ldots,\ell\}$ of nonnegative integers such that $|f(x)-f(y)|\geq p$ if $x$ is a vertex and $y$ is an edge incident to $x$, and $|f(x)-f(y)|\geq 1$ if $x$ and $y$ are a pair of adjacent vertices or a pair of adjacent edges, for all $x$ and $y$ in $V(G)\cup E(G)$. Such a labeling of $G$ is called a $(p,1)$-total labeling of $G$, introduced by Havet and Yu [7, 8]. The $(p,1)$-total labeling number is defined as the minimum value $\ell$, and denoted by $\lambda_{p}^{T}(G)$.

Notice that a $(1,1)$-total labeling of $G$ is equivalent to a total coloring of $G$. Generalizing the Total Coloring Conjecture [2, 13], Havet and Yu [7, 8] conjectured that

holds for any graph $G$, where $\Delta(G)$ denotes the maximum degree of a vertex in $G$. They also showed that (i) $\lambda_{p}^{T}(G)\leq\min\{2\Delta(G)+p-1,\ \chi(G)+\chi^{\prime}(G)+p-2\}$ for any graph $G$ where $\chi(G)$ and $\chi^{\prime}(G)$ denote the chromatic number and the chromatic index of $G$, respectively, (ii) $\lambda_{2}^{T}(G)\leq 2\Delta(G)$ if $p=2$ and $\Delta(G)\geq 2$, (iii) $\lambda_{2}^{T}(G)\leq 2\Delta(G)-1$ if $p=2$ and $\Delta(G)$ is an odd $\geq 5$, and (iv) $\lambda_{p}^{T}(G)\leq n+2p-2$ if $G$ is the complete graph where $n=|V(G)|$; the conjecture (1) is true if (a) $p\geq\Delta(G)$, (b) $p=2$ and $\Delta(G)\leq 3$, or (c) $G$ is the complete graph. By (i), it follows that $\lambda_{p}^{T}(G)\leq\Delta(G)+p$ for any bipartite graph [1, 7, 8] (by $\chi(G)\leq 2$ and $\chi^{\prime}(G)=\Delta(G)$ [9]). Also, Bazzaro et al. [1] investigated that $\lambda_{p}^{T}(G)\leq\Delta(G)+p+s$ for any $s$-degenerated graph (by $\chi(G)\leq s+1$ and $\chi^{\prime}(G)\leq\Delta(G)+1$ [12]), where an $s$-degenerated graph $G$ is a graph which can be reduced to a trivial graph by successive removal of vertices with degree at most $s$, and that $\lambda_{p}^{T}(G)\leq\Delta(G)+p+3$ for any planar graph (by the Four-Color Theorem). They also showed sufficient conditions about $\Delta(G)$ and girth for which the conjecture (1) holds. In [11], Montassier and Raspaud proved that $\lambda_{p}^{T}(G)\leq\Delta(G)+2p-2$ when $p\geq 2$ and $\Delta(G)$ and the maximum average of $G$ degree satisfy some conditions. In [10], Lih et al. showed that $\lambda_{p}^{T}(G)\leq\lfloor 3\Delta(G)/2\rfloor+4p-3$ for any graph $G$. They also investigated $\lambda_{p}^{T}(K_{m,n})$ of the complete bipartite graphs $K_{m,n}$.

Let $G$ be an outerplanar graph. In [1], Bazzaro et al. pointed out that $\lambda_{p}^{T}(G)\leq\Delta(G)+p+1$ for any outerplanar graph other than an odd cycle, since any outerplaner graph is 2-degenerated, and any outerplaner graph other than an odd cycle satisfies $\chi^{\prime}(G)=\Delta(G)$ [5]. In particular, Chen and Wang [4] conjectured that in the case of $p=2$, $\lambda_{2}^{T}(G)\leq\Delta(G)+2$, which is a better bound than the conjecture (1). They also proved that this conjecture is true if (i) $\Delta(G)\geq 5$, (ii) $\Delta(G)=3$ and $G$ is 2-connected, or (iii) $\Delta(G)=4$ and every two faces consisting of three vertices have no vertex in common. In the case of $\Delta(G)=2$ (i.e., $G$ is a path or a cycle), we can easily see that $\lambda^{T}_{2}(G)\leq 4$, since the incidence graph of a path (resp., a cycle) is also a path (resp., a cycle), and the $L(2,1)$-labeling number $\lambda_{2,1}(C_{n})$ for a cycle $C_{n}$ with $n$ vertices is at most 4 [6]. The cases of $\Delta(G)\in\{0,1\}$ are clear. However, the general cases of $\Delta(G)\in\{3,4\}$ were left open. In this paper, we solve Cheng and Wang’s conjecture completely, by showing the remaining cases of $\Delta(G)\in\{3,4\}$. On the other hand, the bound $\Delta(G)+2$ is tight, since there exist infinitely many outerplanar graphs $G$ such that $\lambda_{2}^{T}(G)=\Delta(G)+2$ if $\Delta(G)\geq 2$, as investigated in [4].

The rest of this paper is organized as follows. Section 2 gives basic definitions. In Sections 3 and 4, we show that $\lambda_{2}^{T}(G)\leq\Delta(G)+2$ in the cases of $\Delta(G)=3$ and $\Delta(G)=4$, recpectively.

A graph $G$ is an ordered set of its vertex set $V(G)$ and edge set $E(G)$ and is denoted by $G=(V(G),E(G))$. We assume throughout this paper that a graph is undirected, simple and connected, unless otherwise stated. Therefore, an edge $e\in E(G)$ is an unordered pair of vertices $u$ and $v$, which are end vertices of $e$, and we often denote it by $e=(u,v)$. For two graphs $G_{1}$ and $G_{2}$, we denote $(V(G_{1})\cup V(G_{2}),E(G_{1})\cup E(G_{2}))$ by $G_{1}+G_{2}$. For a vertex set $V^{\prime}\subseteq V(G)$, let $G-V^{\prime}$ be the subgraph of $G$ induced by $V(G)-V^{\prime}$ . For an edge set $E^{\prime}$, let $G-E^{\prime}$ (resp., $G+E^{\prime}$) be the graph $(V,E-E^{\prime})$ (resp., $(V\cup V(E^{\prime}),E\cup E^{\prime})$ where $V(E^{\prime})$ is the set of end vertices of edges in $E^{\prime}$). Two vertices $u$ and $v$ are adjacent if $(u,v)\in E(G)$. Let $N_{G}(v)$ denote the set of neighbors of a vertex $v$ in $G$; $N_{G}(v)=\{u\in V\mid(u,v)\in E(G)\}$. The degree of a vertex $v$ is $|N_{G}(v)|$, and is denoted by $d_{G}(v)$. A vertex $v$ with $d_{G}(v)=k$ is called a $k$-vertex. We use $\Delta(G)$ (resp., $\delta(G)$) to denote the maximum (resp., minimum) degree of a graph $G$. A path $P$ in $G$ is a sequence $v_{1},v_{2},\ldots,v_{\ell}$ of vertices such that $(v_{i},v_{i+1})\in E(G)$ for $i=1,2,\ldots,\ell-1$; we may denote $P$ as $(v_{1},v_{2},\ldots,v_{\ell})$ and let $V(P)=\{v_{1},v_{2},\ldots,v_{\ell}\}$ and $E(P)=\{(v_{1},v_{2}),(v_{2},v_{3}),\ldots,(v_{\ell-1},v_{\ell})\}$. We often drop $G$ in these notations if there are no confusions.

The vertex-connectivity of a graph $G$, denoted by $\kappa(G)$, is defined as the minimum cardinality of a vertex set $V^{\prime}\subseteq V(G)$ such that $G-V^{\prime}$ is disconnected or trivial. A vertex $v$ is called a cut vertex $($of $G)$ if $G-v$ is disconnected. Notice that a graph $G$ has no cut vertex if and only if $\kappa(G)\geq 2$. Two vertices $u$ and $u^{\prime}$ are called 2-connected if they belong to the same connected component in $G-v$ for any cut vertex $v$ of $G$ with $v\notin\{u,u^{\prime}\}$. A subgraph $G^{\prime}$ of $G$ is called a 2-connected component if every two vertices in $G^{\prime}$ are 2-connected and $G^{\prime}$ is maximal to this property. It is not difficult to see that if $\kappa(G)=1$, then there exists a 2-connected component which contains exactly one cut vertex of $G$.

A graph is called planar if it can be drawn in the plane without generating a crossing by two edges. For such a drawing of $G$ in the plane, after omitting the line segments, the plane is divided into regions; such a region is called a face. A face whose vertex set is $\{u_{1},u_{2},\ldots,u_{k}\}$ with $(u_{i},u_{i+1})\in E(G)$, $i=1,2,\ldots,k$ (where $u_{k+1}=u_{1}$) is denoted by $[u_{1}u_{2}\cdots u_{k}]$. We call a face consisting of $k$ vertices a $k$-face.

A planar graph $G$ is called outerplanar if it can be drawn in the plane so that all vertices lie on the boundary of some face. Such a drawing is referred to as an outerplane graph. The face whose boundary contains all vertices of $G$ is called the outer face and all other faces are called inner faces. We call an edge belonging to the boundary of the outer face an outer edge and all other edges inner edges. An inner face is called an endface if its boundary contains exactly one inner edge (note that there exist at least two endfaces if $G$ contains an inner edge).

Let ${\cal L}_{k}=\{0,1,\ldots,k\}$ and $f:V(G)\cup E(G)\to{\cal L}_{k}$ be a (2,1)-total labeling of $G$. For an edge $e=(u,v)\in E(G)$, we may denote $f(e)$ by $f(uv)$. Let $\overline{f}$ denote the labeling such that $\overline{f}(z)=k-f(z)$ for each $z\in V(G)\cup E(G)$. Notice that $\overline{f}$ is also a (2,1)-total labeling of $G$.

Let $G=(V,E)$ be an outerplane graph with $\Delta(G)=3$. Chen and Wang [4] showed that if $\kappa(G)\geq 2$, then $\lambda_{2}^{T}(G)\leq 5$. First review their constructive proof of the case of $\kappa(G)\geq 2$. Assume that on the boundary of its outerface, each 3-vertex (i.e., each vertex which some inner edge is incident to) is numbered $x_{1},x_{2},\ldots,x_{p}$ in clockwise order. Also assume that for $i=1,2,\ldots,p$, each 2-vertex on the path connecting $x_{i}$ and $x_{i+1}$ (on the boundary of its outerface) is numbered $y_{1}^{i},y_{2}^{i},\ldots,y_{q_{i}}^{i}$ in clockwise order, where $x_{p+1}=x_{1}$. A (2,1)-total labeling $f:V\cup E\to{\cal L}_{5}$ can be obtained by the following algorithm, which slightly generalizes Chen and Wang’s method.

Moreover, we can see the following property.

By using this lemma, we can prove that Chen and Wang’s conjecture is true in the case of $\Delta(G)=3$.