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arXiv 2501.07129math.CO
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不含3-环、4-环和6-环的平面图的$(2,4)$-可着色性

$(2,4)$-Colorability of Planar Graphs Excluding $3$-, $4$-, and $6$-Cycles

Pongpat Sittitrai, Wannapol Pimpasalee, Kittikorn Nakprasit

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AI总结：

针对不含3-、4-、6-环的平面图的缺陷2-着色问题，本文在已有研究证明其为$(0,6)$-可着色和$(3,3)$-可着色的基础上，进一步证明这类图也具有$(2,4)$-可着色性。

摘要：

缺陷k-着色是使用颜色1,2,…,k对图的顶点进行的着色，允许相邻顶点共享同一颜色。图$G$的$(d_1,d_2)$-着色是$G$的一种缺陷2-着色，其中染颜色$i$的任意顶点最多有$d_i$个同色相邻顶点，$i\in\{1,2\}$。若图$G$存在$(d_1,d_2)$-着色，则称其为$(d_1,d_2)$-可着色的。Dross与Ochem、Sittitrai与Pimpasalee分别研究了不含3-环、4-环和6-环的平面图的缺陷2-着色，证明这类图分别是$(0,6)$-可着色和$(3,3)$-可着色的。本文证明了这类图同样是$(2,4)$-可着色的。

英文摘要：

A defective $k$-coloring is a coloring on the vertices of a graph using colors $1,2, \dots, k$ such that adjacent vertices may share the same color. A $(d_1,d_2)$-\emph{coloring} of a graph $G$ is a defective $2$-coloring of $G$ such that any vertex colored by color $i$ has at most $d_i$ adjacent vertices of the same color, where $i\in\{1,2\}$. A graph $G$ is said to be $(d_1,d_2)$-\emph{colorable} if it admits a $(d_1,d_2)$-coloring. Defective $2$-coloring in planar graphs without $3$-cycles, $4$-cycles, and $6$-cycles has been investigated by Dross and Ochem, as well as Sittitrai and Pimpasalee. They showed that such graphs are $(0,6)$-colorable and $(3,3)$-colorable, respectively. In this paper, we proved that these graphs are also $(2,4)$-colorable.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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