A note on the 4-girth-thickness of K_{n,n,n}

TL;DR

This paper determines the 4-girth-thickness of complete tripartite graphs and refutes a previous conjecture about the 4-girth-thickness of the complete graph K_{10}, contributing to graph theory knowledge.

Contribution

It provides exact values for the 4-girth-thickness of K_{n,n,n} and disproves a conjecture regarding K_{10}'s 4-girth-thickness.

Findings

4-girth-thickness of K_{n,n,n} is ceiling((n+1)/2) except for K_{1,1,1}

4-girth-thickness of K_{10} is 3, not 4

Disproves Rubio-Montiel's conjecture on K_{10}

Abstract

The $4$-girth-thickness $\theta(4,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least four whose union is $G$. In this paper, we obtain that the 4-girth-thickness of complete tripartite graph $K_{n,n,n}$ is $\big\lceil\frac{n+1}{2}\big\rceil$ except for $\theta(4,K_{1,1,1})=2$. And we also show that the $4$-girth-thickness of the complete graph $K_{10}$ is three which disprove the conjecture $\theta(4,K_{10})=4$ posed by Rubio-Montiel (Ars Math Contemp 14(2) (2018) 319).