Spanning trees and spanning closed walks with small degrees

Abstract

Let G be a graph and let f be a positive integer-valued function on V(G). In this paper, we show that if for all S⊆V(G), ω(G∖S)<∑v∈S(f(v)−2)+2+ω(G[S]), then G has a spanning tree T containing an arbitrary given matching such that for each vertex v, dT(v)≤f(v), where ω(G∖S) denotes the number of components of G∖S and ω(G[S]) denotes the number of components of the induced subgraph G[S] with the vertex set S. This is an improvement of several results. Next, we prove that if for all S⊆V(G), ω(G∖S)≤∑v∈S(f(v)−1)+1, then G admits a spanning closed walk passing through the edges of an arbitrary given matching meeting each vertex v at most f(v) times. This result solves a long-standing conjecture due to Jackson and Wormald (1990).