On Potentially 3-regular graph graphic Sequences

TL;DR

This paper characterizes graphic sequences that can realize a 3-regular subgraph with 6 vertices, specifically focusing on $K_{3,3}$ and $K_6-C_6$, extending previous theorems in graph theory.

Contribution

It provides a complete characterization of potentially $H$-graphic sequences for specific 3-regular graphs with 6 vertices, including $K_{3,3}$ and $K_6-C_6$, advancing understanding of graph realizations.

Findings

Characterization of potentially $K_{3,3}$-graphic sequences.

Characterization of potentially $K_6-C_6$-graphic sequences.

Extension of Yin's theorem on graphic sequences.

Abstract

For given a graph $H$, a graphic sequence $\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\pi$ containing $H$ as a subgraph. In this paper, we characterize the potentially $H$-graphic sequences where $H$ denotes 3-regular graph with 6 vertices. In other words, we characterize the potentially $K_{3,3}$ and $K_6-C_6$-graphic sequences where $K_{r,r}$ is an $r\times r$ complete bipartite graph. One of these characterizations implies a theorem due to Yin [25].