For a graph $G$, let $\tau_{\mathrm{iso}}(G)$ denote the number of isomorphism classes of its spanning trees. For every fixed $d\ge3$ and all sufficiently large $n$, we prove that every connected $n$-vertex graph $G$ with $\delta(G)\ge d$ satisfies \[\tau_{\mathrm{iso}}(G)\ge \tau_{\mathrm{iso}}(K_{d,n-d})=A_dn^{d-1}+O...
Lehel's conjecture states that every 2-edge-colouring of K_n admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by {\L}uczak, R\"odl, and Szemer\'edi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomass\'e in 2010. In this paper, we co...
Pedro Araújo, Xiao-Chuan Liu, Taísa L. Martins et al.· 0 citations
The celebrated result of Johansson, Kahn and Vu determined the threshold order for clique factors in random graphs, and subsequent work identified the sharp threshold and the corresponding hitting-time phenomenon. In this paper we study the probability that there is no $K_r$-factor above the threshold and, more general...
Zhi-Fei Yan· 0 citations
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