Spanning path-cycle systems with given end-vertices in regular graphs (full version)

TL;DR

This paper proves a new theorem on spanning path-cycle systems with specified end-vertices in regular graphs, extending previous results with a different proof approach.

Contribution

It introduces a novel proof for spanning path-cycle decompositions with given end-vertices in regular graphs, generalizing prior work for 3-regular graphs.

Findings

Proves existence of vertex-disjoint paths and cycles covering the entire graph.

Ensures paths connect specified vertices with minimum distance constraints.

Extends previous results to r-regular graphs with r≥4.

Abstract

We prove the following theorem. Let $r\ge 4$ be an integer, and $G$ be a $K_{1,r}$-free $r$-edge-connected $r$-regular graph. Then, for every set $W$ of even number of vertices of $G$ such that the distance between any two vertices of $W$ in $G$ is at least 3, $G$ has vertex-disjoint paths and cycles $P_1, \ldots, P_m, C_1, \ldots, C_n$ such that (i) $V(G)=V(P_1) \cup \cdots \cup V(P_m) \cup V(C_1) \cup \cdots \cup V(C_n)$, (ii) each path $P_i$ connects two vertices of $W$, and (iii) the set of the end-vertices of $P_i$'s is equal to $W$. A similar result for a 3-regular graph is obtained in [Graphs Combin. {\bf 39} (2023) \#85]. However, our proof is widely different from its proof.