论文标题
将多媒体分解为本地不规则的跨文章
On decomposing multigraphs into locally irregular submultigraphs
论文作者
论文摘要
局部不规则的多编码是一个多编码，其相邻顶点具有不同的度。本地不规则的边缘着色是多式$ g $的边缘着色，因此每种颜色都会引起$ g $的本地不规则的临时跨越图。我们说，如果多数式$ g $承认本地不规则的边缘着色，则在本地不规则的色彩，并且我们用$ {\ rm lir}（g）$表示$ g $的本地不规则彩色指数，这是本地不规则不规则的多色$ g $ g $的本地不规则边缘着色所需的最小颜色。我们推测，对于每个连接的图形$ g $，这不是同构至$ k_2 $的，多graph $^2g $从$ g $获得的，每倍的$ g $都可以通过加倍的每个边缘允许$ {\ rm lir}（^2G）\ leq 2 $。这个概念与众所周知的1-2-3猜想，局部不规则猜想，（2，2）猜想和有关边缘着色的其他类似问题。我们显示了该猜想的图表类，例如路径，周期，车轮，完整的图形，完整的$ k $ - 分段图和两部分图。我们还证明，使用我们的两部分图的结果，我们证明了所有2-次数的局部不规则色度指数。
A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. We say that a multigraph $G$ is locally irregular colorable if it admits a locally irregular edge coloring and we denote by ${\rm lir}(G)$ the locally irregular chromatic index of $G$, which is the smallest number of colors required in a locally irregular edge coloring of a locally irregular colorable multigraph $G$. We conjecture that for every connected graph $G$, which is not isomorphic to $K_2$, multigraph $^2G$ obtained from $G$ by doubling each edge admits ${\rm lir}(^2G)\leq 2$. This concept is closely related to the well known 1-2-3 Conjecture, Local Irregularity Conjecture, (2, 2) Conjecture and other similar problems concerning edge colorings. We show this conjecture holds for graph classes like paths, cycles, wheels, complete graphs, complete $k$-partite graphs and bipartite graphs. We also prove the general bound for locally irregular chromatic index for all 2-multigraphs using our result for bipartite graphs.
