Lower bounds for independence numbers of some locally sparse graphs

Yusheng Li and Qizhong Lin ()
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Yusheng Li: Tongji University
Qizhong Lin: Fuzhou University

Journal of Combinatorial Optimization, 2014, vol. 28, issue 4, No 1, 717-725

Abstract: Abstract An $$m$$ -distinct-coloring is a proper vertex-coloring $$c$$ of a graph $$G$$ if for each vertex $$v\in V$$ , any color appears in at most one of $$N_0(v)$$ , $$N_1(v)$$ , $$\ldots $$ , and $$N_m(v)$$ , where $$N_i(v)$$ is the set of vertices at distance $$i$$ from $$v$$ . In this note, we show that if $$G$$ is $$C_{2m+1}$$ -free which is assigned an $$(m+1)$$ -distinct-coloring $$c$$ , then $$\alpha (G)c(G)^{1/m}\ge \Omega \Big (\sum _{v} c(v)^{1/m}\Big )$$ , where $$c(G)$$ is the number of colors used in $$c$$ and $$c(v)$$ is the number of different colors appearing in $$N_1(v)$$ . Moreover, we obtain that if $$G$$ has $$N$$ vertices and it contains neither $$C_{2m+1}$$ nor $$C_{2m}$$ , then $$\alpha (G)\ge \Omega \big ((N\log N)^{m/(m+1)}\big )$$ . The algorithm in the proof for the first result is random, and that for the second is constructive.

Keywords: Independence number; Locally sparse graph; Locally distinct coloring (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10878-012-9578-4

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