Neighbor sum distinguishing total chromatic number of planar graphs

Abstract

Let G = (V(G), E(G)) be a graph and ϕ be a proper k-total coloring of G. Set $f_{\varphi }\left(v\right)={\sum }_{uv\in E\left(G\right)}\varphi \left(uv\right)+\varphi \left(v\right),$ for each v ∈ V(G). If f_ϕ(u) ≠ f_ϕ(v) for each edge uv ∈ E(G), the coloring ϕ is called a k-neighbor sum distinguishing total coloring of G. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by ${\chi }_{\Sigma }^{″}\left(G\right)$. In this paper, by using the famous Combinatorial Nullstellensatz, we determine ${\chi }_{\Sigma }^{″}\left(G\right)$ for any planar graph G with Δ(G) ≥ 13.

Introduction

The terminology and notation used but undefined in this paper can be found in [2]. In this paper, we only consider simple, finite and undirected graphs. Let $G=\left(V\right(G),E(G)$) be a graph with maximum degree Δ(G). Let d_G(v) or simply d(v) denote the degree of a vertex v in G. A vertex v is a $t-vertex$ ($t^{+}-vertex,$ $t^{-}-vertex$) if $d\left(v\right)=t$ (d(v) ≥ t, d(v) ≤ t). Let d_t(v) ($d_{t^{-}}\left(v\right),$ $d_{t^{+}}\left(v\right)$) denote the number of t-vertices ($t^{-}$-vertices, $t^{+}$-vertices) adjacent to v. Let N(v) denote the neighbor set of the vertex v of G. Let $G=\left(V\right(G),E(G),F(G\left)\right)$ be a plane graph. An l-face $f=v_1{v}_2\dots v_l\in F\left(G\right)$ is a ($b_1,b_2,\dots ,b_l$)-face, if v_i is a b_i-vertex, for $i=1,2,\dots ,l$.

Given a graph G and a proper k-total coloring ϕ : $V\left(G\right)\cup E\left(G\right)\to \{1,2,\dots ,k\}$. The coloring ϕ is a k-neighbor sum (set) distinguishing total coloring if f_ϕ(u) ≠ f_ϕ(v)(S_ϕ(u) ≠ S_ϕ(v)) for each edge uv ∈ E(G), where f_ϕ(v)(S_ϕ(v)) is the sum (set) of colors on the edges incident with v and the color on the vertex v. The smallest number k in such a coloring of graph G is the neighbor sum (set) distinguishing total chromatic number, denoted by ${\chi }_{\Sigma }^{″}\left(G\right)$$\left({\chi }_a^{″}\left(G\right)\right)$. Clearly, for any graph G, ${\chi }_a^{″}\left(G\right)\le {\chi }_{\Sigma }^{″}\left(G\right)$. All colorings considered in this paper are proper colorings. Zhang et al. [25] proposed the following conjecture.

Conjecture 1.1 [25]

For any graph G, ${\chi }_a^{″}\left(G\right)\le \Delta \left(G\right)+3$.

Huang and Wang [8] showed that Conjecture 1.1 holds for planar graphs with Δ(G) ≥ 11, which was extended to Δ(G) ≥ 10 by Cheng et al. [4]. Wang and Huang [21] showed that if G is a planar graph with Δ(G) ≥ 13, then ${\chi }_a^{″}\left(G\right)\le \Delta \left(G\right)+2,$ meanwhile, they proved that if G has no adjacent Δ(G)-vertices, then ${\chi }_a^{″}\left(G\right)=\Delta \left(G\right)+1$.

For k-neighbor sum distinguishing total coloring (or simply k-tnsd-coloring), Pilśniak and Woźniak [12] gave the following conjecture.

Conjecture 1.2 [12]

For any graph G, ${\chi }_{\Sigma }^{″}\left(G\right)\le \Delta \left(G\right)+3$.

Since ${\chi }_a^{″}\left(G\right)\le {\chi }_{\Sigma }^{″}\left(G\right)$ for any graph G, Conjecture 1.2 implies Conjecture 1.1. Conjecture 1.2 was confirmed for complete graphs, cycles, bipartite graphs and subcubic graphs in [12]. Dong and Wang [5] showed that Conjecture 1.2 holds for some sparse graphs. Yao et al. [23], [24] considered tnsd-coloring of degenerate graphs. [6], [14] considered the list version of tnsd-coloring of graphs. [7], [10], [11], [17], [18], [19], [20] considered tnsd-coloring of planar graphs with cycle restrictions. Song and Xu [16] determined ${\chi }_{\Sigma }^{″}\left(G\right)$ for any K₄-minor free graph G with Δ(G) ≥ 5. Li et al. [9] showed that Conjecture 1.2 holds for planar graphs with maximum degree at least 13, and subsequently the result was improved by Qu et al. [13] and by Yang et al. [22]. Cheng et al. [3] showed that if G is a planar graph with Δ(G) ≥ 14, then ${\chi }_{\Sigma }^{″}\left(G\right)\le \Delta \left(G\right)+2$. Recently, Song et al. [15] improved this result and get that:

Theorem 1.1 [15]

Let G be a planar graph with maximum degree Δ(G). Then ${\chi }_{\Sigma }^{″}\left(G\right)\le max\{\Delta \left(G\right)+2,14\}$.

In this paper, we get the following results.

Theorem 1.2

Let G be a planar graph without adjacent maximum degree vertices. Then ${\chi }_{\Sigma }^{″}\left(G\right)\le max\{\Delta \left(G\right)+1,14\}$.

Since ${\chi }_{\Sigma }^{″}\left(G\right)\ge \Delta \left(G\right)+1$ for any graph G, and if G has adjacent Δ(G)-vertices, then ${\chi }_{\Sigma }^{″}\left(G\right)\ge \Delta \left(G\right)+2$. Thus we get the following corollary.

Corollary 1.1

Let G be a planar graph and Δ(G) ≥ 13. If G has no adjacent Δ(G)-vertices, then ${\chi }_{\Sigma }^{″}\left(G\right)=\Delta \left(G\right)+1,$ otherwise ${\chi }_{\Sigma }^{″}\left(G\right)=\Delta \left(G\right)+2$.

Clearly, this result implies the result in [21].