论文标题
2种族裔与仙人掌的局部不规则猜想
Local Irregularity Conjecture for 2-multigraphs versus cacti
论文作者
论文摘要
如果每个多层的最终偏见的程度都是不同的，则多编码在局部是不规则的。本地不规则的着色是多式$ g $的边缘着色，因此每种颜色都会引起本地不规则的$ g $的不规则延伸。本地不规则的多色$ g $是任何可以接受本地不规则着色的多编码。我们用$ {\ rm lir}（g）$表示多数$ g $的本地不规则色素索引，这是本地不规则可着色的多数$ g $的本地不规则着色所需的最小颜色。对于图形，定义是相似的。 2个培养书的当地不规则猜想声称，对于每个连接的图形$ g $，这不是同构为$ k_2 $的，从$ g $获得的$ k_2 $，通过加倍的每个边缘来获得的$ g $获得的$ {\ rm lir}（^2G）（^2G）\ leq 2 $。我们显示了仙人掌的猜想。这类图表对于2个传播的局部不规则猜想和本地不规则猜想很重要，该猜想声称每个本地不规则的可着色图$ g $都满足$ {\ rm lir}（g）\ leq 3 $。一开始，已经观察到所有不是局部不规则的可着色图都是仙人掌。最近，已经证明，只有一个仙人掌需要4种颜色才能局部不规则的着色，因此局部不规则猜想被拒绝了。
A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. A locally irregular colorable multigraph $G$ is any multigraph which admits a locally irregular coloring. We denote by ${\rm lir}(G)$ the locally irregular chromatic index of a multigraph $G$, which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph $G$. In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph $G$, which is not isomorphic to $K_2$, multigraph $^2G$ obtained from $G$ by doubling each edge satisfies ${\rm lir}(^2G)\leq 2$. We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph $G$ satisfies ${\rm lir}(G)\leq 3$. At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved.
