For positive integers $n>dgeq k$, let $phi(n,d,k)$ denote the least integer $phi$ such that every $n$-vertex graph with at least $phi$ vertices of degree at least $d$ contains a path on $k+1$ vertices. Many years ago, ErdH{o}s, Faudree, Schelp and Simonovits proposed the study of the function $phi(n,d,k)$, and conjectured that for any positive integers $n>dgeq k$, it holds that $phi(n,d,k)leq lfloorfrac{k-1}{2}rfloorlfloorfrac{n}{d+1}rfloor+epsilon$, where $epsilon=1$ if $k$ is odd and $epsilon=2$ otherwise. In this paper we determine the value of the function $phi(n,d,k)$ exactly. This confirms the above conjecture of ErdH{o}s et al. for all positive integers $k eq 4$ and in a corrected form for the case $k=4$. Our proof utilizes, among others, a lemma of ErdH{o}s et al., a theorem of Jackson, and a (slight) extension of a very recent theorem of Kostochka, Luo and Zirlin, where the latter two results concern maximum cycles in bipartite graphs. Besides, we construct examples to provide answers to two closely related questions raised by ErdH{o}s et al.
تحميل البحث