A note on graph proper total colorings with many distinguishing constraints

doi:10.1016/j.ipl.2015.11.014

Information Processing Letters

Volume 116, Issue 6, June 2016, Pages 396-400

https://doi.org/10.1016/j.ipl.2015.11.014 Get rights and content

Highlights

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Graph coloring theory has a wide range of applications in many scientific fields. As an application, graph distinguishing colorings may be connected with the frequency assignment problem in wireless communication. The vertices (nodes) of graphs (networks) represent transmitters, the set $C [u, f]$ of colors assigned to a vertex u and the edges (links) incident to u under a total coloring f indicates the frequencies usable. One constrain $C [u, f] \neq C [v, f]$ ensures the corresponding two stations u, v can operate on a wide range of frequencies without the danger of interfering with each other. We propose new total colorings having more constraints for researching deeply the frequency assignment problem. Under a proper total coloring f of a simple graph G, $f (u)$ and $f (u v)$ are the color assigned to a vertex u and the color assigned to an edge uv, respectively. We use $C (f, u)$ to denote the set of colors assigned to the edges incident to u, $C 〈 f, u 〉$ the union of $f (u)$ and the set of colors assigned to the neighbors of u, so we have other two color sets $C [f, u] = C (f, u) \cup {f (u)}$ , and $C_{2} [f, u] = C (f, u) \cup C 〈 f, u 〉$ . We say f an adjacent vertex distinguishing total coloring (AVDTC) of G if one constraint $C [f, u] \neq C [f, v]$ holds for each edge uv of G, and the minimum number of k colors required for which G admits an AVDTC is denoted as $χ_{a s}^{″} (G)$ , which is related with a conjecture: $χ_{a s}^{″} (G) \leq Δ (G) + 3$ , where $Δ (G)$ is the maximum degree. We call f a 4-adjacent vertex distinguishing total coloring (4-AVDTC) of G if four distinguishing constraints $C (f, x) \neq C (f, y)$ , $C 〈 f, x 〉 \neq C 〈 f, y 〉$ , $C [f, x] \neq C [f, y]$ and $C_{2} [f, x] \neq C_{2} [f, y]$ hold simultaneously true for every edge uv of G, and the least number of k colors required for which G admits a k-4-AVDTC is denoted by $χ_{4 a s}^{″} (G)$ . We conjecture $χ_{4 a s}^{″} (G) \leq Δ (G) + 4$ if any edge uv of G holds that the set of neighbors of the vertex u differs from the set of neighbors of the vertex v.

Abstract

A proper edge-coloring of a simple graph G is called a vertex distinguishing edge-coloring if for any two distinct vertices u and v of G, the set of the colors assigned to the edges incident to u differs from the set of the colors assigned to the edges incident to v. We extend such distinguishing edge colorings to proper total colorings with many distinguishing constraints and color several kinds of graphs totally with the least number of colors such that the graphs admit proper total colorings having at least four distinguishing constraints.

Section snippets

Introduction and concepts

Labeled graphs are becoming an increasingly useful family of mathematical models for a broad range of applications, such as time tabling and scheduling, frequency assignment, register allocation, computer security and so on. In [2], Burris and Schelp introduced a proper edge-coloring of a simple graph G that is called a vertex distinguishing edge-coloring (vdec) if for any two distinct vertices u and v of G, the set of the colors assigned to the edges incident to u differs from the set of the

Graphs having 8-vdtcs or 4-avdtcs

Lemma 1

A graph $G \in F_{3 s} (n)$ admits a total coloring f with $C 〈 f, u 〉 \neq C 〈 f, v 〉$ for distinct vertices $u, v \in V (G)$ (resp. for every edge $u v \in E (G)$ ) if and only if $N (u) \cup {u} \neq N (v) \cup {v}$ for distinct $u, v \in V (G)$ (resp. every edge $u v \in E (G)$ ).

Proof

To show the proof of ‘if’, we take a total coloring f with $C 〈 f, u 〉 \neq C 〈 f, v 〉$ for distinct vertices $u, v \in V (G)$ . If $u v \in E (G)$ , $C 〈 f, u 〉 \neq C 〈 f, v 〉$ means that ${f (x) : x \in N (u)} ∖ {f (u), f (v)} \neq {f (x) : x \in N (v)} ∖ {f (u), f (v)}$ , and furthermore $N (u) \cup {u} \neq N (v) \cup {v}$ . If $u v \notin E (G)$ , $C 〈 f, u 〉 \neq C 〈 f, v 〉$ means $N (u) = N (v)$ , or $N (u) \neq N (v)$ . No

Acknowledgements

The authors wish to sincerely thank the referees for their valuable and thoughtful suggestions, which greatly improve the present paper. The author, Bing Yao, was supported by the National Natural Science Foundation of China under No. 61163054 and No. 61363060. The author, Han Ren, was supported by the National Natural Science Foundation of China under No. 11171114.

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