Graphs with Strong Proper Connection Numbers and Large Cliques

Abstract

In this paper, we mainly investigate graphs with a small (strong) proper connection number and a large clique number. First, we discuss the (strong) proper connection number of a graph G of order n and ω(G)=n−i for 1⩽i⩽3. Next, we investigate the rainbow connection number of a graph G of order n, diam(G)≥3 and ω(G)=n−i for 2⩽i⩽3.