Optimal r-dynamic coloring of sparse graphs

Dan Yi, Junlei Zhu (), Lixia Feng, Jiaxin Wang and Mengyini Yang
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Dan Yi: Jiaxing University
Junlei Zhu: Jiaxing University
Lixia Feng: Jiaxing University
Jiaxin Wang: Jiaxing University
Mengyini Yang: Jiaxing University

Journal of Combinatorial Optimization, 2019, vol. 38, issue 2, No 11, 545-555

Abstract: Abstract An r-dynamick-coloring of a graphG is a proper k-coloring such that every vertex v in V(G) has neighbors in at least $$min\{d(v),r\}$$ m i n { d ( v ) , r } different classes. The r-dynamic chromatic number ofG, written $$\chi _{r}(G)$$ χ r ( G ) , is the minimum integer k such that G has such a coloring. In this paper, we investigate the r-dynamic $$(r+1)$$ ( r + 1 ) -coloring (i.e. optimal r-dynamic coloring) of sparse graphs and prove that $$\chi _{r}(G)\le r+1$$ χ r ( G ) ≤ r + 1 holds if G is a planar graph with $$g(G)\ge 7$$ g ( G ) ≥ 7 and $$r\ge 16$$ r ≥ 16 , which is a generalization of the case $$r=\Delta $$ r = Δ .

Keywords: Girth; Sparse graph; r-Dynamic coloring (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10878-019-00387-0

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