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-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every edge-colored complete graph without containing monochromatic triangles always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the excepti…
We extend results of Gyárfás and Füredi on the largest monochromatic component in $r$-colored complete $k$-uniform hypergraphs to the setting of random hypergraphs. We also study long monochromatic loose cycles in $r$-colored random hypergraphs. In particular, we obtain a random analog of a result of Gyárfás, Sárközy, and Szemerédi on the longest monochromatic loose cycle in $2$-colored complete $k$-uniform hypergraphs.
A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the same component of $G$. Following Łuczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings.
An $r$-edge coloring of a graph or hypergraph $G=(V,E)$ is a map $c:E\to \{0, \dots, r-1\}$. Extending results of Rado and answering questions of Rado, Gyárfás and Sárközy we prove that (1.) the vertex set of every $r$-edge colored countably infinite complete $k$-uniform hypergraph can be partitioned into $r$ monochromatic tight paths with distinct colors (a tight path in a $k$-uniform hypergraph is a sequence of distinct vertices such that every set of $k$ consecutive vertices forms an edge), (2.) for all natural numbers $r$ and $k$ there is a natural number $M$ such that the vertex set of ev…
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 $δ^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with…
Gyárfás and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of $K_{n,n}$ there exists a monochromatic path on at least $2\lceil n/2\rceil$ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, if $G$ is a balanced bipartite graph on $2n$ vertices with minimum degree at least $(3/4+o(1))n$, then in every 2-coloring of the edges of $G$, either there exists a monochromatic cycle on at least $(1+o(1))n$ vertices, or the coloring of $G$ is close to an extremal coloring -- in which c…
We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices. This implies a generalization for large $n$ of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ the…
A Gallai $k$-coloring is a $k$-edge coloring of a complete graph in which there are no rainbow triangles. For two given graphs $H, G$ and two positive integers $k,s$ with that $s\leq k$, the $k$-colored Gallai-Ramsey number $gr_{k}(K_{3}: s\cdot H,~ (k-s)\cdot G)$ is the minimum integer $n$ such that every Gallai $k$-colored $K_{n}$ contains a monochromatic copy of $H$ colored by one of the first $s$ colors or a monochromatic copy of $G$ colored by one of the remaining $k-s$ colors. In this paper, we determine the value of Gallai-Ramsey number in the case that $H=K_{4}^{+}$ and $G=K_{3}$. Thus…
For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. For a connected graph $G$, the monochromatic disconnection number, denoted by $md(G)$, of $G$ is the maximum number of colors that are needed in order to make $G$ monochromatically disconnected. We will show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus-Gaddum-type results for $md…
A digraph is {\em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $d\geq 2$, 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 $d\geq 2$, proving $4\leq f(2)\le …
We show that for any finite coloring of the group $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ and for any positive integer $k$, there always exists a monochromatic regular $k$-gon in $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ with respect to the word length metric induced by the standard generating set; the edge length of which is estimated.
For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. The monochromatic disconnection number, denoted by $md(G)$, of a connected graph $G$ is the maximum number of colors that are allowed to make $G$ monochromatically disconnected. In this paper, we solve the Erdős-Gallai-type problems for the monochromatic disconnection, and give the monochromatic disconnection numbers for four graph produ…
It is known that in any $r$-coloring of the edges of a complete $r$-uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on $n$ vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3-coloring of the edges? Gyárfás proved that $(2n+3)/3$ is an absolute lower bound and that this lower bound is best possible for infinitely many $n$. On the other hand, we prove that for almost all Steiner triple systems the lower bound is actually $(1-o(1))n$. We obtain this result as a consequence of a more general theorem which show…
A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. In this paper, we study two kinds of Erdős-Gallai-type problems for $tmc(G)$ and completely solve th…
Let $T_{n}$ be an arc-colored tournament of order $n$. The maximum monochromatic indegree $Δ^{-mon}(T_{n})$ (resp. outdegree $Δ^{+mon}(T_{n})$) of $T_{n}$ is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of $T_{n}$. The irregularity $i(T_{n})$ of $T_{n}$ is the maximum difference between the indegree and outdegree of a vertex of $T_{n}$. A subdigraph $H$ of an arc-colored digraph $D$ is called rainbow if each pair of arcs in $H$ have distinct colors. In this paper, we show that each vertex $v$ in an arc-colored tournament $T_{n}$ with $Δ^{-mon}(T_n)\leqΔ^{…
Hindman proved that, whenever the set $\mathbb{N}$ of naturals is finitely colored, there must exist non-constant monochromatic solution of the equation $a+b=cd$. In this paper we extend this result for dense subsemigroups of $((0, \infty), +)$ to near zero.
A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are connected by at least $k$ edge-disjoint monochromatic paths, and $G$ is said to be uniformly monochromatic $k$-edge-connected if every two distinct vertices are connected by at least $k$ edge-disjoint monochromatic paths such that all edges of these $k$ paths colored with a same color. We use $mc_k(G)$ and $umc_k(G)$ to denote the maximum number of colors th…
Erdős, Gyárfás, and Pyber (1991) conjectured that every $r$-colored complete graph can be partitioned into at most $r-1$ monochromatic components; this is a strengthening of a conjecture of Lovász (1975) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into $r$ monochromatic components is possible for sufficiently large $r$-colored complete graphs. We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random gr…
Let G be a vertex-colored graph. A vertex cut S of G is called a monochromatic vertex cut if the vertices of S are colored with the same color. A graph G is monochromatically vertex-disconnected if any two nonadjacent vertices of G has a monochromatic vertex cut separating them. The monochromatic vertex-disconnection number of G, denoted by mvd(G), is the maximum number of colors that are used to make G monochromatically vertex-disconnected. In this paper, the connection between the graph parameters are studied: mvd(G), connectivity and block decomposition. We determine the value of mvd(G) for…
A graph $G$ arrows a graph $H$ if in every $2$-edge-coloring of $G$ there exists a monochromatic copy of $H$. Schelp had the idea that if the complete graph $K_n$ arrows a small graph $H$, then every "dense" subgraph of $K_n$ also arrows $H$, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large $n$, if $n = 3t+r$ where $r \in \{0,1,2\}$ and $G$ is an $n$-vertex graph with $δ(G) \ge (3n-1)/4$, then for every $2$-edge-coloring of $G$, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same color, or …
A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A path in a total-colored graph is a {\it total monochromatic path} if all the edges and internal vertices on the path have the same color. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a total monochromatic path of the graph. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TM…
Let $T(H, G_n)$ be the number of monochromatic copies of a fixed connected graph $H$ in a uniformly random coloring of the vertices of the graph $G_n$. In this paper we give a complete characterization of the limiting distribution of $T(H, G_n)$, when $\{G_n\}_{n \geq 1}$ is a converging sequence of dense graphs. When the number of colors grows to infinity, depending on whether the expected value remains bounded, $T(H, G_n)$ either converges to a finite linear combination of independent Poisson variables or a normal distribution. On the other hand, when the number of colors is fixed, $T(H, G_n…
Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component, or a green matching of size at least $m$ in a component. In this paper, we determine the exact value of $r(k,l,m)$ completely. Applying a technique originated by Łuczak that applies Szemerédi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obtain th…
Let G be a vertex-colored graph. We call a vertex cut S of G a monochromatic vertex cut if the vertices of S are colored with the same color. The graph G is monochromatically vertex-disconnected if any two nonadjacent vertices of G has a monochromatic vertex cut separating them. The monochromatic vertex-disconnection number of G, denoted by mvd(G), is the maximum number of colors that are used to make G monochromatically vertex-disconnected. In this paper, we propose an algorithm to compute mvd(G) and give an mvd-coloring of the graph G. We run this algorithm with an example written in Java. T…
Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if a binary matroid of rank $r$ is colored with exactly $r$ colors, then $M$ either contains a rainbow colored circuit or a monochromatic cut. As the class of binary matroids is closed under taking duals, this immediately implies that if $M$ is colored with exactly $n-r$ colors, then $M$ either contains a rainbow colored cut or a monochromatic circuit. As a byproduct, we give a characterization of binary matroids in terms of …
A tree $T$ in an edge-colored graph $H$ is called a \emph{monochromatic tree} if all the edges of $T$ have the same color. For $S\subseteq V(H)$, a \emph{monochromatic $S$-tree} in $H$ is a monochromatic tree of $H$ containing the vertices of $S$. For a connected graph $G$ and a given integer $k$ with $2\leq k\leq |V(G)|$, the \emph{$k$-monochromatic index $mx_k(G)$} of $G$ is the maximum number of colors needed such that for each subset $S\subseteq V(G)$ of $k$ vertices, there exists a monochromatic $S$-tree. In this paper, we prove that for any connected graph $G$, $mx_k(G)=|E(G)|-|V(G)|+2$ …
A Gallai $k$-coloring is a $k$-edge coloring of a complete graph in which there are no rainbow triangles. For given graphs $G_1, G_2, G_3$ and nonnegative integers $r, s, t$ with that $k=r+s+t$, the $k$-colored Gallai-Ramsey number $gr_{k}(K_{3}: r\cdot G_1,~ s\cdot G_2, ~t\cdot G_3)$ is the minimum integer $n$ such that every Gallai $k$-colored $K_{n}$ contains a monochromatic copy of $G_1$ colored by one of the first $r$ colors or a monochromatic copy of $G_2$ colored by one of the middle $s$ colors or a monochromatic copy of $G_3$ colored by one of the last $t$ colors. In this paper, we det…
Gyárfás conjectured in 2011 that every $r$-edge-colored $K_n$ contains a monochromatic component of bounded ("perhaps three") diameter on at least $n/(r-1)$ vertices. Letzter proved this conjecture with diameter four. In this note we improve the result in the case of $r=3$: We show that in every $3$-edge-coloring of $K_n$ either there is a monochromatic component of diameter at most three on at least $n/2$ vertices or every color class is spanning and has diameter at most four.
For all positive integers $r\geq 3$ and $n$ such that $r^2-r$ divides $n$ and an affine plane of order $r$ exists, we construct an $r$-edge colored graph with minimum degree $(1-\frac{r-2}{r^2-r})n-2$ such that the largest monochromatic component has order less than $\frac{n}{r-1}$. This generalizes an example of Guggiari and Scott and, independently, Rahimi for $r=3$ and thus disproves a conjecture of Gyárfás and Sárközy for all integers $r\geq 3$ such that an affine plane of order $r$ exists.
A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices on the path have the same color. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. Note that a TMC-coloring does not exist if $G$ is not connected, in which case we simply…
It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{n,n}$ there is a monochromatic connected component with at least ${2n\over r}$ vertices. It would be interesting to know whether we can additionally require that this large component be balanced; that is, is it true that in every $r$-coloring of $K_{n,n}$ there is a monochromatic component that meets both sides in at least $n/r$ vertices? Over forty years ago, Gyárfás and Lehel and independently Faudree and Schelp proved that any $2$-colored $K_{n,n}$ contains a monochromatic $P_n$. Very recently, Buci…
Erdős and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n) vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed color patterns ensures larger monochromatic cliques. Specifically, it claims that for any fixed integer k and any clique K on k vertices edge-colored with two colors, there is a positive constant a such that in any complete n-vertex graph edge-colored with two colors that does not contain a copy of K, there is a monochromatic cli…
Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider monochromatic bipartite graphs since the numbers are known to grow more slowly than for non-bipartite graphs. The main result shows that it suffices to consider only $3$-colorings which have a special partition of the vertices. Using this tool, we find several sharp numbers and conjecture the sharp value for all bipartite graphs. In particular, we determine the Gal…
An open conjecture of Erdős states that for every positive integer $k$ there is a (least) positive integer $f(k)$ so that whenever a tournament has its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$ vertices so that every vertex has a monochromatic path to some point in $S$. We consider a related question and show that for every (finite or infinite) cardinal $κ>0$ there is a cardinal $ λ_κ$ such that in every $κ$-edge-coloured tournament there exist disjoint vertex sets $K,S$ with total size at most $ λ_κ$ so that every vertex $ v $ has a monochromatic path of length a…
A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call $k$ paths $P_1,\cdots,P_k$ rainbow monochromatic paths if every $P_i$ is monochromatic and for any two $i\neq j$, $P_i$ and $P_j$ have different colors. An edge-coloring of a graph $G$ is said to be a rainbow monochromatic $k$-edge-connection coloring (or $RMC_k$-coloring for short) if every two distinct vertices of $G$ are connected by at least $k$ rainbow monochromatic paths. We use $rmc_k(G)$ to denote the maximum number of colors that ensures $G$ has an $RMC_k$-coloring, and t…
An edge-colored graph $G$ is called properly colored if no two adjacent edges share a color in $G$. An edge-colored connected graph $G$ is called properly connected if between every pair of distinct vertices, there exists a path that is properly colored. In this paper, we discuss how to make a connected graph properly connected efficiently. More precisely, we consider the problem to convert a given monochromatic graph into properly connected by recoloring $p$ edges with $q$ colors so that $p+q$ is as small as possible. We discuss how this can be done efficiently for some restricted graphs, suc…
