Acyclic Choosability of Graphs with Bounded Degree

Wang, Juan; Miao, Lian Ying; Li, Jin Bo; Liu, Yunlong

doi:10.1007/s10114-022-0097-7

Cited by 1 publication

(1 citation statement)

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“…Wood [10] studied the star and acyclic chromatic numbers of subdivision graph G ′ of a graph G. A great deal of research has been conducted since then. Recently, Wang et al [9] studied the acyclic choosability of graphs with bounded degrees. Acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of Hessians using automatic differentiation or finite differencing when both sparsity and symmetry are exploited.…”

Section: Introductionmentioning

confidence: 99%

Acyclic and Star Coloring of Powers of Paths and Cycles

Etawi

Ghanem²,

Al-Ezeh³

2022

Eur. J. Pure Appl. Math.

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Let $G=(V,E)$ be a graph. The $k^{th}-$ power of $G$ denoted by $G^{k}$ is the graph whose vertex set is $V$ and in which two vertices are adjacent if and only if their distance in $G$ is at most $k.$ A vertex coloring of $G$ is acyclic if each bichromatic subgraph is a forest. A star coloring of $G$ is an acyclic coloring in which each bichromatic subgraph is a star forest.The minimum number of colors such that $G$ admits an acyclic (star) coloring is called the acyclic (star) chromatic number of G, and denoted by $\chi_{a}(G)(\chi_{s}(G))$. In this paper we prove that for $n\geq k+1,$ $\chi_{s}(P_{n}^{k})=\min\{\lfloor\frac{k+n+1}{2}\rfloor,2k+1\}$ and $\chi_{a}(P_{n}^{k})=k+1.$ Further we show that for $n\geq(k+1)^{2}$, $%2k+1\leq\chi _{s}(C_{n}^{k})\leq2k+2$ and $k+2\leq\chi_{a}(C_{n}^{k})\leq k+3.$ Finally, we derive the formula $\chi_{a}(C_{n}^{k})=k+2$ for $% n\geq(k+1)^{3}.$

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Section: Introductionmentioning

confidence: 99%