We prove that for any partition of the plane into a closed set $C$ and an open set $O$ and for any configuration $T$ of three points, there is a translated and rotated copy of $T$ contained in $C$ or in $O$. Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition and any configuration $T$ which is a vertex set of a non-equilateral triangle there is a copy of $T$ contained in the interior of one of the two partition classes. Furthermore, we give the characterization of these "polygonal" partitions that avoid copies of a given equilateral triple. These results support a conjecture of Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, which states that every two-coloring of the plane contains a monochromatic copy of any nonequilateral triple of points; on the other hand, we disprove a stronger conjecture by the same authors, by providing non-trivial examples of two-colorings that avoid a given equilateral triple.
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…
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 $t_3(K(n,n))=3.$
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.
Let $K_n$ be the complete graph with $n$ vertices and $c_1, c_2, ..., c_r$ be $r$ different colors. Suppose we randomly and uniformly color the edges of $K_n$ in $c_1, c_2, ..., c_r$. Then we get a random graph, denoted by $\mathcal{K}_n^r$. In the paper, we investigate the asymptotic properties of several kinds of monochromatic and heterochromatic subgraphs in $\mathcal{K}_n^r$. Accurate threshold functions in some cases are also obtained.
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…
A $d$-dimensional grid is a set of the form $R = [a_1] \times ... \times [a_d]$. A $d$-dimensional box is a set of the form $\{b_1,c_1\} \times ... \times \{b_d,c_d\}$. When a grid is $c$-colored, must it admit a monochromatic box? If so, we say that $R$ is $c$-guaranteed. This question is a relaxation of one attack on bounding the van der Waerden numbers, and also arises as a natural hypergraph Ramsey problem (viz. the Ramsey numbers of hyperoctahedra). We give conditions on the $a_i$ for $R$ to be $c$-guaranteed that are asymptotically tight, and analyze the set of minimally $c$-guaranteed g…
We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2α(G)$ monochromatic cycles, where $α(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1…
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.
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…
It has been conjectured that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every (r - 1)-coloring of the edges of the complete r-uniform hypergraph on n vertices. In this paper, we show that the statement of this conjecture is true with r-2 colors (instead of r-1 colors) by showing that there is a monochromatic Hamiltonian t-tight Berge-cycle in every b r-2 / t-1 -edge coloring of Kr n for any fixed r > t >= 2 and sufficiently large n. Also, we give a proof for this conjecture when r = 4 (the first open case). These results improve the previously…
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…
In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.
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…
In the problem of 2-coloring without monochromatic triangles (or triangle-tree 2-coloring), vertices of the simple, connected, undirected graph are colored with either 'black' or 'white' such that there are no 3 mutually adjacent vertices of the same color. In this paper we are positively answering the question posed in our previous work, namely, if there exists an algorithm solving 2-coloring without monochromatic triangles on planar graphs with linear-time complexity.
We prove that in every $2$-edge-colouring of $K_n$ there is a collection of $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, thus confirming a conjecture of Erdős. We also prove a corresponding stability result, showing that $2$-colourings that are close to attaining the aforementioned bound have a colour class which is close to bipartite. As part of our proof, we confirm a recent conjecture of Tyomkyn about the fractional version of this problem.
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…
We give a full, correct proof of the following result, earlier claimed by Erdős and Komjáth. If the Continuum Hypothesis holds then there is a coloring of the plane with countably many colors, with no monocolored right triangle.
Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$. We prove that whenever $|E| + \min\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$. We conjecture a natural generalization to partitions with more blocks.
Let $χ_Δ(\mathbb{R}^{n})$ denote the minimum number of colors needed to color $\mathbb{R}^{n}$ so that there will not be a monochromatic equilateral triangle with side length $1$. Using the slice rank method, we reprove a result of Frankl and Rodl, and show that $χ_Δ\left(\mathbb{R}^{n}\right)$ grows exponentially with $n$. This technique substantially improves upon the best known quantitative lower bounds for $χ_Δ\left(\mathbb{R}^{n}\right)$, and we obtain \[ χ_Δ\left(\mathbb{R}^{n}\right)>(1.01446+o(1))^{n}. \]
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 …
A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} (MC-coloring, for short) if there is a monochromatic path joining any two vertices in $G$. The \emph{monochromatic connection number}, denoted by $mc(G)$, is defined to be the maximum number of colors used in an MC-coloring of a graph $G$. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we will study two kinds of Erdős-Gallai-type problems for $mc(G)$, and …
On the Number of Almost Empty Monochromatic Triangles
University of Pennsylvania（宾夕法尼亚大学）；National University of Singapore（新加坡国立大学）；Indian Statistical Institute, Kolkata（印度统计研究所，加尔各答）；University of Michigan（密歇根大学）
Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is defined as the number of colors of the edges incident to $v$. The color number of $G$ is defined as the number of colors of the edges in $G$. A rainbow triangle is one in which every pair of edges have distinct colors. In this paper we give some sufficient conditions for the existence of rainbow triangles in edge-colored graphs in terms of color degree, color number and edge number. As a corollary, a conjecture proposed by Li and Wang (Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin…
Let $D$ be an arc-colored digraph. The arc number $a(D)$ of $D$ is defined as the number of arcs of $D$. The color number $c(D)$ of $D$ is defined as the number of colors assigned to the arcs of $D$. A rainbow triangle in $D$ is a directed triangle in which every pair of arcs have distinct colors. Let $f(D)$ be the smallest integer such that if $c(D)\geq f(D)$, then $D$ contains a rainbow triangle. In this paper we obtain $f(\overleftrightarrow{K}_{n})$ and $f(T_n)$, where $\overleftrightarrow{K}_{n}$ is a complete digraph of order $n$ and $T_n$ is a strongly connected tournament of order $n$.…
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…
For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\geq \binom{n+1}{2}+k-1$ for a non-negative integer $k$, then $G$ contains at least $k$ rainbow triangles. For $n\geq 3k$, we show that this result is best possible, and we completely characterize the class of edge-colored graphs for which this result is sharp. Furthermore, we show that an edge-colored graph $G$ contains at least $k$ rainbow triangles if $\sum\limits_{v\in V(G)} d^c_G(v)\geq …
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…
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^{…
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…
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 …
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $δ^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $δ^c(G)\geq \frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $δ^c(G)\geq \frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k…
For positive integers $c, s \geq 1$, let $M_3(c, s)$ be the least integer such that any set of at least $M_3(c, s)$ points in the plane, no three on a line and colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. The case $s=0$, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that $M_3(1, 0)=3$, $M_3(2, 0)=9$ and $M_3(c, 0)=\infty$, for $c\geq 3$. In this paper we extend these results when $c \geq 2$ and $s \geq 1$. We prove that the least integer $λ_3(c)$ such that $M_3…
Erdos and Sos proposed a problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a)+ F(b)+F(c)+F(d)+abc+abd+acd+bcd, where a+b+c+d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4k for all k. These results imply that lim F(n) n^3/6 = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.
Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\hat c=\hat c(n)$ with $0<c<\hat c<C$ such that for any $\eps > 0$, as $n$ tends to infinity $$Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0$$ and $$Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1.$$ A crucial tool that is used in the proof an…
We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literature. We propose the new structural parameter for undirected, simple graphs -- the triangle-free chromatic number $χ_3$. We bound $χ_3$ by other known structural parameters. We also present two classes of g…
We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles under certain conditions, then it requires at least seven colors. This is also true for a coloring using uniformly colored polygons if it has a point bordering at least four polygons.
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…
Institut für Mathematik, Technische Universität Berlin（柏林工业大学数学研究所）；HUN-REN Alfréd Rényi Institute of Mathematics and ELTE Eötvös Loránd University（匈牙利科学院阿尔弗雷德·雷尼数学研究所和布达佩斯罗兰大学）
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 …
The flip operation on colored inner-triangle-free triangulations of a convex polygon is studied. It is shown that the affine Weyl group $\widetilde{C}_n$ acts transitively on these triangulations by colored flips, and that the resulting colored flip graph is closely related to a lower interval in the weak order on $\widetilde{C}_n$. Lattice properties of this order are then applied to compute the diameter.
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…
