论文标题
仙人掌的局部不规则边缘着色的注释
A note on the locally irregular edge colorings of cacti
论文作者
论文摘要
如果每个边缘的最终偏见的程度都不同，则图是局部不规则的。如果每种颜色诱导G的局部不规则亚图。可着色图G的局部不规则色索引X'irr（g）是G的局部不规则边缘着色所需的最小颜色。局部不规则性猜想声称，所有可着色图最多都需要3种颜色，以使本地不规则的边缘着色。最近，已经观察到猜想不适合弓形图B，因为B可着色，并且需要至少4种颜色才能为局部不规则的边缘着色。由于B是仙人掌图，并且所有不可分析的图也都是仙人掌，因此这似乎是局部不规则猜想的相关图。在本文中，我们确定所有可色彩仙人掌图的x'irr（g）<= 4。
A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed that the conjecture does not hold for the bow-tie graph B, since B is colorable and requires at least 4 colors for a locally irregular edge coloring. Since B is a cactus graph and all non-colorable graphs are also cacti, this seems to be a relevant class of graphs for the Local Irregularity Conjecture. In this paper we establish that X'irr(G)<= 4 for all colorable cactus graphs.
