The 4-girth-thickness of the complete multipartite graph

TL;DR

This paper determines the 4-girth-thickness of complete m-partite graphs with even-sized parts, providing exact values for a specific graph parameter related to planar subgraphs with girth constraints.

Contribution

The paper introduces the exact calculation of the 4-girth-thickness for complete m-partite graphs with even parts, advancing understanding of graph decomposition under girth constraints.

Findings

Calculated the 4-girth-thickness for complete m-partite graphs with even parts.

Provided explicit formulas or values for this girth-thickness.

Enhanced the theoretical framework for graph decomposition with girth restrictions.

Abstract

The $g$-girth-thickness $\theta(g,G)$ 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 $4$-girth-thickness $\theta(4,G)$ of the complete $m$-partite graph $G$ when each part has an even number of vertices.