论文标题
Mc-numbers对平面图的极端图和平面图的分类
Extremal graphs and classification of planar graphs by MC-numbers
论文作者
论文摘要
连接图$ g $的边缘颜色称为{\ em单色连接着色}（简称为mc-coloring），如果任何两个$ g $的顶点均由$ g $中的单色路径连接。对于连接的图形$ g $，{\ em单色连接编号}（简称为$ g $的mc-number），用$ mc（g）$表示，是确保$ g $具有单色连接着色的最大颜色数量。这个概念是由Caro and Yuster在2011年引入的。他们证明了$ MC（G）\ Leq M-N+K $如果$ G $不是$ K $连接的图。在本文中，我们用$ mc（g）= m-n+k+1 $和$ mc（g）= m-n+k $描绘所有图，如果$ g $是$ k $连接但不是$（k+1）$连接的图。我们还证明，如果$ g $是平面图，则$ mc（g）\ leq m-n+4 $ 4 $，并将所有平面图按其单色连接数进行分类。
An edge-coloring of a connected graph $G$ is called a {\em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {\em monochromatic connection number} (MC-number for short) of $G$, denoted by $mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic connection coloring by using this number of colors. This concept was introduced by Caro and Yuster in 2011. They proved that $mc(G)\leq m-n+k$ if $G$ is not a $k$-connected graph. In this paper we depict all graphs with $mc(G)=m-n+k+1$ and $mc(G)= m-n+k$ if $G$ is a $k$-connected but not $(k+1)$-connected graph. We also prove that $mc(G)\leq m-n+4$ if $G$ is a planar graph, and classify all planar graphs by their monochromatic connectivity numbers.
