arXiv subjects

Rachel Galindo

Publications and source records attributed to Rachel Galindo.

2 recordsLinked to original sources

arXiv2025

Cliques and High Odd Holes in Graphs with Chromatic Number Equal to Maximum Degree

We give a uniform and self-contained proof that if $G$ is a connected graph with $\chi(G) = \Delta(G)$ and $G\neq \overline{C_7}$, then $G$ contains either $K_{\Delta(G)}$ or an odd hole where every vertex has degree at least $\Delta(G)-1$ in $G$. This was previously proved in series of two papers by Chen, Lan, Lin, and Zhou, who used the Strong Perfect Graph Theorem for the cases $\Delta(G)=4, 5, 6$.

math.CO↗

arXiv2024

On graphs with chromatic number and maximum degree both equal to nine

An equivalent version of the Borodin-Kostochka Conjecture, due to Cranston and Rabern, says that any graph with $\chi = \Delta = 9$ contains $K_3 \lor E_6$ as a subgraph. Here we prove several results in support of this conjecture, where vertex-criticality and forbidden substructure conditions get us either close or all the way to containing $K_3 \lor E_6$.

math.CO↗