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Spanning trees and spanning closed walks with small degrees

Hasanvand, M ; Sharif University of Technology | 2022

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2022.112998
  3. Publisher: Elsevier B.V , 2022
  4. Abstract:
  5. 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). © 2022 Elsevier B.V
  6. Keywords:
  7. Connected factor ; Matching ; Spanning closed walk ; Spanning tree ; Toughness
  8. Source: Discrete Mathematics ; Volume 345, Issue 10 , 2022 ; 0012365X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0012365X22002047