图因子的Kahn–Lovász型不等式
Kahn--Lovász-type inequalities for graph factors
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中文总结 AI 辅助
本文将Kahn–Lovász定理推广到F-因子情形，得到F-因子的渐近精确上界，还证明其多重图类似结果，进而得到一类连通图F的Kruskal–Katona型界。
中文摘要 AI 辅助
Kahn–Lovász定理根据图的度序列给出了图中完美匹配数的精确上界，推广了二分图的经典Brégman–Minc不等式。本文中，我们对每个哈密顿图F，建立了Kahn–Lovász定理的渐近精确推广，即F-因子的相关结果。作为推论，我们渐近确定了n个顶点、m条边的图中F-因子的最大数量，得到了Kruskal–Katona型定理的F-因子类似结果。我们还证明了Kahn–Lovász定理的多重图类似结果，结合哈密顿图的相关结果，得到了另一类连通图F的渐近精确Kruskal–Katona型界，这类F包括由两个顶点不交、长度相等的圈构成且其并集覆盖V(F)的图。
英文摘要
The Kahn--Lovász theorem gives a sharp upper bound on the number of perfect matchings in a graph in terms of its degree sequence, extending the classical Brégman--Minc inequality for bipartite graphs. In this paper, we establish an asymptotically sharp extension of the Kahn--Lovász theorem to $F$-factors for every Hamiltonian graph $F$. As a consequence, we asymptotically determine the maximum number of $F$-factors in an $n$-vertex $m$-edge graph, yielding an $F$-factor analogue of Kruskal--Katona-type theorems. We also prove a multigraph analogue of the Kahn--Lovász theorem. Combining this with our results for Hamiltonian graphs, we obtain an asymptotically sharp Kruskal--Katona-type bound for a further class of connected graphs $F$, including those containing two vertex-disjoint cycles of equal length whose union spans $V(F)$.
