INJECTIVELY DELTA CHOOSABLE GRAPHS

Title & Authors
INJECTIVELY DELTA CHOOSABLE GRAPHS
Kim, Seog-Jin; Park, Won-Jin;

Abstract
An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors. A graph G is said to be injectively $\small{k}$-choosable if any list $\small{L(v)}$ of size at least $\small{k}$ for every vertex $\small{v}$ allows an injective coloring $\small{{\phi}(v)}$ such that $\small{{\phi}(v){\in}L(v)}$ for every $\small{v{\in}V(G)}$. The least $\small{k}$ for which G is injectively $\small{k}$-choosable is the injective choosability number of G, denoted by $\small{{\chi}^l_i(G)}$. In this paper, we obtain new sufficient conditions to be $\small{{\chi}^l_i(G)={\Delta}(G)}$. Maximum average degree, mad(G), is defined by mad(G) = max{2e(H)/n(H) : H is a subgraph of G}. We prove that if mad(G) < $\small{\frac{8k-3}{3k}}$, then $\small{{\chi}^l_i(G)={\Delta}(G)}$ where $\small{k={\Delta}(G)}$ and $\small{{\Delta}(G){\geq}6}$. In addition, when $\small{{\Delta}(G)=5}$ we prove that $\small{{\chi}^l_i(G)={\Delta}(G)}$ if mad(G) < $\small{\frac{17}{7}}$, and when $\small{{\Delta}(G)=4}$ we prove that $\small{{\chi}^l_i(G)={\Delta}(G)}$ if mad(G) < $\small{\frac{7}{3}}$. These results generalize some of previous results in [1, 4].
Keywords
injective coloring;list coloring;maximum average degree;discharging;
Language
English
Cited by
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