Fengnan Yanling | Search arXiv

arXiv2015

Proper connection number and graph products

A path $P$ in an edge-colored graph $G$ is called \emph{a proper path} if no two adjacent edges of $P$ are colored the same, and $G$ is \emph{proper connected} if every two vertices of $G$ are connected by a proper path in $G$. The \emph{proper connection number} of a connected graph $G$, denoted by $pc(G)$, is the minimum number of colors that are needed to make $G$ proper connected. In this paper, we study the proper connection number on the lexicographical, strong, Cartesian, and direct product and present several upper bounds for these products of graphs.

math.CO↗

arXiv2015

Monochromatic connectivity and graph products

The concept of monochromatic connectivity was introduced by Caro and Yuster. A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} ($MC$-coloring, for short) if there is a monochromatic path joining any two vertices in $G$. The \emph{monochromatic connection number}, denoted by $mc(G)$, is defined to be the maximum number of colors used in an $MC$-coloring of a graph $G$. In this paper, we study the monochromatic connection number on the lexicographical, strong, Cartesian and direct product and present several upper and lower bounds for these products of graphs.

math.CO↗