2021/12/30 by Zilong Yan, Yan, Zilong, Yuejian Peng +1 · 1 citation
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Advanced Topology and Set Theory #Advanced Graph Theory Research
paper · pdf · doi:10.48550/arxiv.2112.14955
We say that a graph F can be embedded into a graph G if G contains an isomorphic copy of F as a subgraph. Guo and Volkmann \citeGV conjectured that if G is a connected graph with at least n vertices and minimum degree at least n-3, then any tree with n vertices and maximum degree at most n-4 can be embedded into G. In this paper, we give a result slightly stronger than this conjecture and obtain a sufficient and necessary condition that a tree with n vertices and maximum degree at most n-3 can be embedded into a connected graph G with at least n vertices and minimum degree at least n-3. Our result implies that the conjecture of Guo and Volkmann is true with one exception. We also give an application to the Ramsey number of a tree versus a star.