The 6-girth-thickness of the complete graph

TL;DR

This paper determines the 6-girth-thickness of complete graphs for most cases and computes the missing values for the 4-girth-thickness using computer assistance, advancing understanding of graph decompositions.

Contribution

It provides the first comprehensive determination of the 6-girth-thickness of complete graphs and fills gaps in the 4-girth-thickness data through computational methods.

Findings

Calculated the 6-girth-thickness of $K_n$ for almost all cases.

Computed missing values of the 4-girth-thickness of $K_n$ using computer algorithms.

Enhanced understanding of planar subgraph decompositions with girth constraints.

Abstract

The $g$-girth-thickness $\theta(g,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least $g$ whose union is $G$. In this paper, we determine the $6$-girth-thickness $\theta(6,K_n)$ of the complete graph $K_n$ in almost all cases. And also, we calculate by computer the missing value of $\theta(4,K_n)$.