Cycles and paths through vertices whose degrees are at least the bipartite-hole-number

TL;DR

This paper extends classical results by showing that in 2-connected graphs, cycles can be found passing through all vertices with degrees above the bipartite-hole-number, and paths connecting high-degree vertices also contain all such vertices.

Contribution

It introduces new theorems linking the bipartite-hole-number to the existence of specific cycles and paths involving high-degree vertices in graphs.

Findings

Existence of cycles passing through all vertices with degrees at least the bipartite-hole-number in 2-connected graphs.

Existence of paths connecting vertices with degrees above the bipartite-hole-number, containing all such vertices.

Extension of classical degree-based cycle and path results using the bipartite-hole-number concept.

Abstract

The bipartite-hole-number of a graph $G$, denoted by $\widetilde{\alpha}(G)$, is the minimum integer $k$ such that there exist positive integers $s$ and $t$ with $s + t = k + 1$, satisfying the property that for any two disjoint sets $A, B \subseteq V(G)$ with $|A| = s$ and $|B| = t$, there is at least one edge between $A$ and $B$. In 1992, Bollob\'as and Brightwell, and independently Shi, proved that every $2$-connected graph of order $n$ contains a cycle passing through all vertices whose degrees are at least $\frac{n}{2}$. Motivated by their result, we show that in any $2$-connected graph of order $n$, there exists a cycle containing all vertices whose degrees are at least $\widetilde{\alpha}(G)$. Moreover, we prove that for any pair of vertices in a connected graph $G$, if their degrees are at least $\widetilde{\alpha}(G) + 1$, then there exists a path joining them that contains all vertices whose degrees are at least $\widetilde{\alpha}(G) + 1$. The results extend two existing ones.