A note on edge degree and spanning trail containing given edges

TL;DR

This paper investigates conditions under which a simple graph contains a spanning closed trail including a specified set of edges, revealing structural properties and implications for line graphs' Hamiltonicity.

Contribution

It establishes a new sufficient condition involving edge degrees for the existence of spanning closed trails containing given edges, and characterizes the structure of graphs that do not meet this condition.

Findings

Graphs with degree sum condition contain spanning closed trails for small edge sets.

Line graphs of such graphs are proven to be k-Hamiltonian.

Characterization of exceptional graphs lacking the spanning trail property.

Abstract

Let $G$ be a simple graph with $n\geq4$ vertices and $d(x)+d(y)\geq n+k$ for each edge $xy\in E(G)$. In this work we prove that $G$ either contains a spanning closed trail containing any given edge set $X$ if $|X|\leq k$, or $G$ is a well characterized graph. As a corollary, we show that line graphs of such graphs are $k$-hamiltonian.