Related papers: Rainbow eulerian multidigraphs and the product of …
We consider coloring problems in the distributed message-passing setting. The previously-known deterministic algorithms for edge-coloring employed at least (2Delta - 1) colors, even though any graph admits an edge-coloring with Delta + 1…
We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Omega(n^{1/3}log^2 n) which uses at most two colors, and this bound is tight up to a constant factor. This…
An acyclic digraph in which every vertex has indegree at most $i$ and outdegree at most $j$ is called an $(i,j)$ digraph for some positive integers $i$ and $j$. The phylogeny graph of a digraph $D$ has $V(D)$ as the vertex set and an edge…
We first give an alternative proof of the Alon-Tarsi list coloring theorem. We use the ideas from this proof to obtain the following result, which is an additive coloring analog of the Alon-Tarsi Theorem: Let $G$ be a graph and let $D$ be…
In a recent paper by a superset of the authors it was proved that for every primitive 3-constrained space $\Gamma$ of finite diameter $\delta$ from Cherlin's catalogue of metrically homogeneous graphs, there exists a finite family $\mathcal…
In this paper we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular we ask how many colors and how many random edges are needed so…
We prove the following conjecture of S. Thomass\'e: for every (potentially infinite) digraph $ D $ it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation $ D^{*} $ of $ D $…
The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the…
Given a multi-hypergraph $G$ that is edge-colored into color classes $E_1, \ldots, E_n$, a full rainbow matching is a matching of $G$ that contains exactly one edge from each color class $E_i$. One way to guarantee the existence of a full…
An edge colored graph is said to contain rainbow-$F$ if $F$ is a subgraph and every edge receives a different color. In 2007, Keevash, Mubayi, Sudakov, and Verstra\"ete introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a…
We generalize the well known exchange property of Coxeter groups to the setting of edge-colored graphs. This work aims to unify and extend the results of our companion article, "odd Verma's theorem", which were originally established for…
An edge coloring of a graph $G$ is to color all the edges in the graph such that adjacent edges receive different colors. It is acyclic if each cycle in the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon, Sudakov and…
Let $G$ be an edge-colored graph on $n$ vertices. The minimum color degree of $G$, denoted by $\delta^c(G)$, is defined as the minimum number of colors assigned to the edges incident to a vertex in $G$. In 2013, H. Li proved that an…
Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring…
Let $H_{n,p,r}^{(k)}$ denote a randomly colored random hypergraph, constructed on the vertex set $[n]$ by taking each $k$-tuple independently with probability $p$, and then independently coloring it with a random color from the set $[r]$.…
In this paper, we consider a variant of dichromatic number on digraphs with prescribed sets of arcs. Let $D$ be a digraph and let $Z_1, Z_2$ be two sets of arcs in $D$. For a subdigraph $H$ of $D$, let $A(H)$ denote the set of all arcs of…
Motivated to find the answers to some of the questions that have occurred in recent papers dealing with Hamiltonian cycles (abbreviated HCs) in some special classes of grid graphs we started the investigation of spanning unions of cycles,…
The automorphism group of the composition of graphs $G \circ H$ contains the wreath product $Aut(H) \wr Aut(G)$ of the automorphism groups of the corresponding graphs. The classical problem considered by Sabidussi and Hemminger was under…
Let $\Delta(G)$ and $\chi'(G)$ be the maximum degree and chromatic index of a graph $G$, respectively. Appearing in different forms, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) made the following conjecture: Every…
A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for $n$ and $C$ large enough, if $G$ is an edge-colored copy of $K_n$ in which each color…