Packing Independent Cliques in $K_4$-minor-free Graphs

TL;DR

This paper proves that in $K_4$-minor-free and subcubic graphs, the maximum ratio of independent cliques in an indeque set is 1/2, settling two conjectures and advancing understanding of graph clique packings.

Contribution

It establishes the indeque ratio of $K_4$-minor-free graphs and subcubic graphs as 1/2, confirming two conjectures in the field.

Findings

Indeque ratio of $K_4$-minor-free graphs is 1/2.

Indeque ratio of subcubic graphs is 1/2.

Settles two conjectures by Biro, Collado, and Zamora.

Abstract

Let $G$ be a graph and $S$ be a set of cliques of $G$. The set $S$ is an indeque set if every component of $G[S]$, the subgraph induced by vertices of $S$, is a clique. In this paper, we prove that the indeque ratio of $K_4$-minor-free graphs is $\frac 1 2$, which settle two conjectures of Biro, Collado and Zamora. We also show that the indeque ratio of subcubic graphs is $\frac 1 2$.