6 research outputs found
A Survey on Monochromatic Connections of Graphs
The concept of monochromatic connection of graphs was introduced by Caro and
Yuster in 2011. Recently, a lot of results have been published about it. In
this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following
categories: monochromatic connection coloring of edge-version, monochromatic
connection coloring of vertex-version, monochromatic index, monochromatic
connection coloring of total-version.Comment: 26 pages, 3 figure
Erdős-Gallai-Type Results for Total Monochromatic Connection of Graphs
A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a total monochromatically-connecting coloring (TMC-coloring, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph G, the total monochromatic connection number, denoted by tmc(G), is defined as the maximum number of colors used in a TMC-coloring of G. In this paper, we study two kinds of Erdős-Gallai-type problems for tmc(G) and completely solve them