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

Benjamin Xiao; Dong Ye

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

Combinatorics 2025-12-23 v1

Authors: Benjamin Xiao , Dong Ye

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 $\frac 1 2$ .

Keywords

graph theory clique packing $K_4$-minor-free indeque set graph coloring combinatorics extremal graph theory independent sets

Cite

@article{arxiv.2512.18090,
  title  = {Packing Independent Cliques in $K_4$-minor-free Graphs},
  author = {Benjamin Xiao and Dong Ye},
  journal= {arXiv preprint arXiv:2512.18090},
  year   = {2025}
}

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