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- Publisher
- Springer Journals
- Copyright
- Copyright © 2016 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
- Subject
- Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
- ISSN
- 0168-9673
- eISSN
- 1618-3932
- DOI
- 10.1007/s10255-016-0583-8
- Publisher site
- See Article on Publisher Site
Abstract
Let G = (V,E) be a graph and ϕ: V ∪E → {1, 2, · · ·, k} be a total-k-coloring of G. Let f(v)(S(v)) denote the sum(set) of the color of vertex v and the colors of the edges incident with v. The total coloring ϕ is called neighbor sum distinguishing if (f(u) ≠ f(v)) for each edge uv ∈ E(G). We say that ϕ is neighbor set distinguishing or adjacent vertex distinguishing if S(u) ≠ S(v) for each edge uv ∈ E(G). For both problems, we have conjectures that such colorings exist for any graph G if k ≥ Δ(G) + 3. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs, which is denoted by mad (G). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that these two conjectures hold for sparse graphs in their list versions. More precisely, we prove that every graph G with maximum degree Δ(G) and maximum average degree mad(G) has ch Σ ″ (G) ≤ Δ(G) + 3 (where ch Σ ″ (G) is the neighbor sum distinguishing total choice number of G) if there exists a pair $$(k,m) \in \{ (6,4),(5,\tfrac{{18}} {5}),(4,\tfrac{{16}} {5})\}$$ such that Δ(G) ≥ k and mad (G) <m.
Journal
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Apr 29, 2016