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A digraph is {em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $dgeq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete digraph (including loops) can be covered by the vertices of at most $f(d)$ monochromatic $d$-dominating subgraphs. Note that the existence of $f(d)$ is not obvious -- indeed, the question which motivated this paper was simply to determine whether $f(d)$ is bounded, even for $d=2$. We answer this question affirmatively for all $dgeq 2$, proving $4leq f(2)le 8$ and $2dleq f(d)le 2dleft(frac{d^{d}-1}{d-1}right)$ for all $dge 3$. We also give an example to show that there is no analogous bound for more than two colors. Our result provides a positive answer to a question regarding an infinite analogue of the Burr-ErdH{o}s conjecture on the Ramsey numbers of $d$-degenerate graphs. Moreover, a special case of our result is related to properties of $d$-paradoxical tournaments.
Kostochka and Thomason independently showed that any graph with average degree $Omega(rsqrt{log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $Omega(rsqrt{log r})$ contains a $K_r$ minor, a partial result towards Hadwige
We prove a conjecture of Fox, Huang, and Lee that characterizes directed graphs that have constant density in all tournaments: they are disjoint unions of trees that are each constructed in a certain recursive way.
Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs are shown
In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $delta^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $vin V(G)$. A cycle of $(G,c)$ is called proper if any t
The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoi