The 4-girth-thickness of the complete multipartite graph

Abstract

The g-girth-thickness θ(g, G) of a graph G is the smallest number of planar subgraphs of girth at least g whose union is G. In this paper, we calculate the 4-girththickness θ(4, G) of the complete m-partite graph G when each part has an even number of vertices.

The 6-girth-thickness of the complete graph

“…The concept of the thickness was introduced by Tutte [19]. The problem to determine the thickness of a graph G is NP-hard [15], and only a few of exact results are known, for instance, when G is a complete graph [2,5,6], a complete multipartite graph [7,11,18,21,22] or a hypercube [14].…”

The 4-girth-thickness of the complete graph

“…Exact results also are known when g > 3 and finite, for instance, the 4-girth-thickness of the complete graph [9,11,18], the 4-girth-thickness of the complete multipartite graph [11,19] and the 6-girth-thickness of the complete graph [9]. Owing to the fact that the hypercube and the complete bipartite are triangle-free graphs, their thickness equal their 4-girth-thickness which were calculate in [15] and partially calculate in [7,13], respectively.…”