Improved upper bound for the degenerate and star chromatic numbers of graphs

Jiansheng Cai (), Xueliang Li and Guiying Yan
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Jiansheng Cai: Weifang University
Xueliang Li: Nankai University
Guiying Yan: Chinese Academy of Sciences

Journal of Combinatorial Optimization, 2017, vol. 34, issue 2, No 10, 452 pages

Abstract: Abstract Let $$G=G(V,E)$$ G = G ( V , E ) be a graph. A proper coloring of G is a function $$f:V\rightarrow N$$ f : V → N such that $$f(x)\ne f(y)$$ f ( x ) ≠ f ( y ) for every edge $$xy\in E$$ x y ∈ E . A proper coloring of a graph G such that for every $$k\ge 1$$ k ≥ 1 , the union of any k color classes induces a $$(k-1)$$ ( k - 1 ) -degenerate subgraph is called a degenerate coloring; a proper coloring of a graph with no two-colored $$P_{4}$$ P 4 is called a star coloring. If a coloring is both degenerate and star, then we call it a degenerate star coloring of graph. The corresponding chromatic number is denoted as $$\chi _{sd}(G)$$ χ s d ( G ) . In this paper, we employ entropy compression method to obtain a new upper bound $$\chi _{sd}(G)\le \lceil \frac{19}{6}\Delta ^{\frac{3}{2}}+5\Delta \rceil $$ χ s d ( G ) ≤ ⌈ 19 6 Δ 3 2 + 5 Δ ⌉ for general graph G.

Keywords: Degenerate coloring; Star coloring; Chromatic number; Entropy compression method; Upper bound (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-016-0076-y

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