Multiplicity of Stars in Complete Graphs and Complete \(r\)-partite Graphs

Abstract

Let \(K_r\) be the complete graph on \(r\) vertices in which there exists an edge between every pair of vertices, \(K_{m,n}\) be the complete bipartite graph with \(m\) vertices in one partition and \(n\) vertices in the other partition, where each vertex in one partition is adjacent to each vertex in the other partition, and \(K(n, r)\) be the complete \(r\)-partite graph \(K_{n,n,…,n}\) where each partition has \(n\) vertices. In this paper, we determine the minimum number of monochromatic stars \(K_{1,p}\), \( \forall p \geq 2\), in any \(t\)-coloring (\(t \geq 2\)) of edges of \(K_r\), \(K_{m,n}\), and \(K(n, r)\). Also, we prove that these lower bounds are sharp for all values of \(m, n, p, r\), and \(t\) by giving explicit constructions.