Some new sufficient conditions for $2p$-Hamilton-biconnectedness of graphs

TL;DR

This paper establishes new degree and spectral conditions ensuring $2p$-Hamilton-biconnectedness in bipartite graphs, extending previous results and including nearly balanced cases, with specific bounds and exceptions.

Contribution

It introduces novel sufficient conditions based on degree, edge count, and spectral properties for $2p$-Hamilton-biconnectedness in bipartite graphs, including nearly balanced cases.

Findings

Proves degree and edge conditions for $2p$-Hamilton-biconnectedness.

Provides spectral conditions for $2p$-Hamilton-biconnectedness.

Extends results to nearly balanced bipartite graphs.

Abstract

A balanced bipartite graph $G$ is said to be $2p$-Hamilton-biconnected if for any balanced subset $W$ of size $2p$ of $V(G)$, the subgraph induced by $V(G)\backslash W$ is Hamilton-biconnected. In this paper, we prove that "Let $p\geq0$ and $G$ be a balanced bipartite graph of order $2n$ with minimum degree $\delta(G)\geq k$, where $n\geq 2k-p+2$ and $k\geq p$. If the number of edges $ e(G)>n(n-k+p-1)+(k+2)(k-p+1), $ then $G$ is $2p$-Hamilton-biconnected except some exceptions." Furthermore, this result is used to present two new spectral conditions for a graph to $2p$-Hamilton-biconnected. Moreover, the similar results are also presented for nearly balanced bipartite graphs.