Partitions of graphs and multigraphs under degree constraints

Abstract

Let G be a graph which may have multiple edges but no loops, let μ(G) be the multiplicity of G (which is the maximum among the numbers of edges between a pair of vertices), and let s and t be two nonnegative integers. Let K4− be the graph obtained from K4 by removing an edge, for i∈{1,2}, let Fi be the simple graph obtained from C5 by adding a path of length i joining two nonadjacent vertices of C5. Let F={K2,3,K4−,F1,F2}, and let H be the family of simple graphs obtained from K4− by adding a new vertex and joining it to two distinct vertices of K4−. In this paper, we show that a graph G admits a partition (S,T) such that δ(G[S])≥s and δ(G[T])≥t if, (1) s≥2, t≥2, and G is a simple graph with minimum degree at least s+t−1 and without elements of F as subgraphs; or (2) s≥1, t≥1, G has order at least 5, minimum degree at least s+t+2μ(G)−2, and no elements of H as subgraphs. These improve a few earlier results.