Neighbor sum distinguishing total chromatic number of planar graphs
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 ) be a graph with maximum degree Δ(G). Let dG(v) or simply d(v) denote the degree of a vertex v in G. A vertex v is a ( ) if (d(v) ≥ t, d(v) ≤ t). Let dt(v) ( ) denote the number of t-vertices (-vertices, -vertices) adjacent to v. Let N(v) denote the neighbor set of the vertex v of G. Let be a plane graph. An l-face is a ()-face, if vi is a bi-vertex, for .
Given a graph G and a proper k-total coloring ϕ : . The coloring ϕ is a k-neighbor sum (set) distinguishing total coloring if fϕ(u) ≠ fϕ(v)(Sϕ(u) ≠ Sϕ(v)) for each edge uv ∈ E(G), where fϕ(v)(Sϕ(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 . Clearly, for any graph G, . 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, .
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 meanwhile, they proved that if G has no adjacent Δ(G)-vertices, then .
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, .
Since 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 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 . 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 .
In this paper, we get the following results.
Theorem 1.2 Let G be a planar graph without adjacent maximum degree vertices. Then .
Since for any graph G, and if G has adjacent Δ(G)-vertices, then . 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 otherwise .
Clearly, this result implies the result in [21].
Section snippets
Preliminaries
For any graph G, set for each positive integer. A graph G′ is smaller than G if one of the following holds:
- (1)
|E(G′)| < |E(G)|;
- (2)
and precedes n1(G)) with respect to the standard lexicographic order, where .
A graph is minimum for a property when no smaller graph satisfies it.
Let be a polynomial in n (n ≥ 1) variables. Set be the coefficient of the monomial
The proof of Theorem 1.2
We will prove Theorem 1.2 by contradiction. Let G be a minimum counterexample of Theorem 1.2. Let k = max . By the choice of G, any planar graph G′ without adjacent Δ(G′)-vertices which is smaller than G has a k-tnsd-coloring ϕ′. In the following, we will choose some planar graph G′ smaller than G and extend the coloring ϕ′ of G′ to the desired coloring ϕ of G to obtain a contradiction. Without special remark, for any x ∈ (V(G) ∪ E(G)) ∩ (V(G′) ∪ E(G′)), let . For a vertex v
Acknowledgments
We would like to thank the referees for their valuable comments. This work was supported by the National Natural Science Foundation of China (11671232), the Natural Science Foundation of Hebei Province (A2015202301), the University Science and Technology Project of Hebei Province (ZD2015106, QN2017044).
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