The Monochromatic Connectivity of Graphs

Abstract

In 2011, Caro et al. introduced the monochromatic connection of graphs. An edge-coloring of a connected graph G is called a monochromatically connecting (MC-coloring, for short) if there is a monochromatic path joining any two vertices. The monochromatic connection number mc(G) of a graph G is the maximum integer k such that there is a k-edge-coloring, which is an MC-coloring of G. Clearly, a monochromatic spanning tree can monochromatically connect any two vertices. So for a graph G of order n and size m, mc(G) ≥ m − n + 2. Caro et al. proved that both triangle-free graphs and graphs of diameter at least three meet the lower bound.In this paper, we consider the monochromatic connectivity of graphs containing triangles which meet the lower bound too. Also, in order to study the graphs of diameter two, we present the formula for the monochromatic connectivity of join graphs. This will be helpful to solve the problem for graphs of diameter two.

Extremal graphs and classification of planar graphs by MC-numbers

“…In [8,9], the authors generalized the concept of MC-coloring. For more knowledge about the monochromatic connection of graphs, we refer to [1,3,5,6,10,11]. In [4], Caro and Yuster showed that the bound of the second result is sharp, and they studied wheel graphs, outerplanar graphs and planar graphs with minimum degree three.…”

Algorithm for monochromatic vertex-disconnection of graphs

“…The notion monochromatic vertex-connection coloring was introduced by Cai, Li and Wu [5] in 2018, which is defined from the vertex-version. For more results on the monochromatic connection coloring and vertex-connection coloring, we refer to [4,6,8,14].…”

A Survey on Monochromatic Connections of Graphs

“…[18] Let G be a connected graph of order n ≥ 7. If G does not have subgraphs isomorphic to K − 4 , then mc(G) = m − n + 2, where K − 4 denotes the graph obtained from K 4 by deleting an edge.Theorem 2.3.…”

“…[18] Let G be a connected graph of order n ≥ 7. If G does not have two vertex-disjoint triangles, then mc(G) = m − n + 2.…”

“…18 [16]. Let f (n) be a function satisfying 1 ≤ f (n) < 1 2 n(n − 1) + n. ln log n ≤ f (n) < 1 2 n(n − 1) + n, where l ∈ R + , log n n if f (n) = o(n log n).…”