Complementary Graphs with Flows Less Than Three

TL;DR

This paper improves a graph theory result by showing that for graphs with at least 32 vertices and minimum degree at least 4, either the graph or its complement has a flow index less than 3, using new methods.

Contribution

It introduces a new closure operation and contraction method to strengthen previous flow results for graphs and their complements.

Findings

Graphs with at least 32 vertices and minimum degree ≥ 4 have a flow index less than 3 in either the graph or its complement.

The new methods improve the vertex count bound from 44 to 32 for the flow property.

The results extend understanding of flow properties in complementary graphs.

Abstract

X. Hou, H.-J. Lai, P. Li and C.-Q. Zhang [J. Graph Theory 69 (2012) 464-470] showed that for a simple graph $G$ with $|V(G)|\ge 44$, if $\min\{\delta(G),\delta(G^c)\}\ge 4$, then either $G$ or its complementary graph $G^c$ has a nowhere-zero $3$-flow. In this paper, we improve this result by showing that if $|V(G)|\ge 32$ and $\min\{\delta(G),\delta(G^c)\}\ge 4$, then either $G$ or $G^c$ has flow index strictly less than $3$. Our result is proved by a newly developed closure operation and contraction method.