On the 4-girth-thickness of the line graph of the complete graph

TL;DR

This paper determines the 4-girth-thickness of the line graph of a complete graph for even n and explores its embedding on the projective plane, contributing to graph decomposition and embedding theory.

Contribution

It provides the exact 4-girth-thickness of line graphs of complete graphs for even n and analyzes their embeddability on the projective plane.

Findings

Calculated $ heta(4,L(K_n))$ for even n

Established minimum subgraphs with girth ≥ 4 on the projective plane

Extended understanding of graph girth-thickness and embeddings

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 note, we give the $4$-girth-thickness $\theta(4,L(K_n))$ of the line graph of the complete graph $L(K_n)$ when $n$ is even. We also give the minimum number of subgraphs of $L(K_n)$, which are of girth at least $4$ and embeddable on the projective plane, whose union is $L(K_n)$.