Degree Conditions on Copies of Forests in Graphs

Abstract

For a graph G, let $\mu (G)=\min \{\max \{{d}(x),{d}(y)\}:x\ne y,xy\notin E(G), x,y\in V(G)\}$ if G is non-complete, otherwise, $\mu (G)=+\infty .$ For a given positive number s,  we call that a graph G satisfies Fan-type conditions if $\mu (G)\geqslant s.$ Suppose $\mu (G)\geqslant s,$ then a vertex v is called a small vertex if the degree of v in G is less than s. In this paper, we prove that for a forest F with m edges, if G is a graph of order $n\geqslant |F|$ and $\mu (G)\geqslant m$ with at most $\max \{n-2m,0\}$ small vertices, then G contains a copy of F. We also give examples to illustrate both the bounds in our result are best possible.