ﻻ يوجد ملخص باللغة العربية
For a collection $mathbf{G}={G_1,dots, G_s}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $mathbf{G}$-transversal if there exists a bijection $phi:E(H)rightarrow [s]$ such that $ein E(G_{phi(e)})$ for all $ein E(H)$. We prove that for $|V|=sgeq 3$ and $delta(G_i)geq s/2$ for each $iin [s]$, there exists a $mathbf{G}$-transversal that is a Hamilton cycle. This confirms a conjecture of Aharoni. We also prove an analogous result for perfect matchings.
We prove a `resilience version of Diracs theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $varepsilon>0$, a.a.s. the following holds: let $G$ be any subgraph of the random $
We give the following extension of Baranys colorful Caratheodory theorem: Let M be an oriented matroid and N a matroid with rank function r, both defined on the same ground set V and satisfying rank(M) < rank(N). If every subset A of V with r(V - A)
We say that the families $mathcal F_1,ldots, mathcal F_{s+1}$ of $k$-element subsets of $[n]$ are cross-dependent if there are no pairwise disjoint sets $F_1,ldots, F_{s+1}$, where $F_iin mathcal F_i$ for each $i$. The rainbow version of the ErdH os
A fundamental result of Kuhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H-tiling. We prove a de
For a fixed graph $F$ and an integer $t$, the dfn{rainbow saturation number} of $F$, denoted by $sat_t(n,mathfrak{R}(F))$, is defined as the minimum number of edges in a $t$-edge-colored graph on $n$ vertices which does not contain a dfn{rainbow copy