Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. For $n > k > d$ let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.
This paper investigates vertex colorings of graphs such that some rainbow subgraph~$R$ and some monochromatic subgraph $M$ are forbidden. Previous work focussed on the case that $R=M$. Here we consider the more general case, especially the case that $M=K_2$.
We investigate the distributions of the number of: (1) monochromatic complete subgraphs over edgewise 2-colorings of complete graphs; and (2) monochromatic arithmetic progressions over 2-colorings of intervals, as statistical Ramsey theory questions. We present convincing evidence that both distributions are very well-approximated by the Delaporte distribution.
Given a coloring of the edges of the complete graph on n vertices in k colors, by considering the neighbors of an arbitrary vertex it follows that there is a monochromatic diameter two subgraph on at least 1+(n-1)/k vertices. We show that for $k \ge 3$ this is asymptotically best possible, and that for k=2 there is always a monochromatic diameter two subgraph on at least $\lceil {3 \over 4} n \rceil$ vertices, which again, is best possible.
A well-known result by Haxell and Kohayakawa states that the vertices of an $r$-coloured complete graph can be partitioned into $r$ monochromatic connected subgraphs of distinct colours; this is a slightly weaker variant of a conjecture by Erdős, Pyber and Gyárfás that states that there exists a partition into $r-1$ monochromatic connected subgraphs. We consider a variant of this problem, where the complete graph is replaced by a graph with large minimum degree, and prove two conjectures of Bal and DeBiasio, for two and three colours.
We consider the following question of Bollobas: given an r-colouring of the edges of the complete graph on n vertices, how large a k-connected subgraph can we find using only one colour? We solve this problem asymptotically when r-1 is a prime power, and exactly for 2- and 3-colourings.
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.
Let ${\cal{F}}=\{F_1,F_2,\ldots\}$ be a sequence of graphs such that $F_n$ is a graph on $n$ vertices with maximum degree at most $Δ$. We show that there exists an absolute constant $C$ such that the vertices of any 2-edge-colored complete graph can be partitioned into at most $2^{CΔ\logΔ}$ vertex disjoint monochromatic copies of graphs from ${\cal{F}}$. If each $F_n$ is bipartite, then we can improve this bound to $2^{C Δ}$; this result is optimal up to the constant $C$.
Let $G$ be a $2$-coloring of a complete graph on $n$ vertices, for sufficiently large $n$. We prove that $G$ contains at least $n^{(\frac{1}{4} - o(1))\log n}$ monochromatic complete subgraphs of size $r$, where \[ 0.3\log n < r < 0.7\log n. \] The previously known lower bound on the total number of monochromatic complete subgraphs, due to Székely was $n^{0.1576\log n}$. We also prove that $G$ contains at least $n^{\frac{1}{7} \log n} $ monochromatic complete subgraphs of size $\frac{1}{2}\log n$. If furthermore one assumes that the largest monochromatic complete subgraph in $G$ is of size $(\…
Let $K_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a $2$-colouring of the edges of $K_{\mathbb{N}}$ in which every monochromatic path has density~$0$. However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every $(r+1)$-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length $\ell_i$ for any colour $i\leq r$ can be covered by $\prod_{i\leq r} \ell_i$ pairwise disjoint monochr…
Let $n, k, m$ be positive integers with $n\gg m\gg k$, and let $\mathcal{A}$ be the set of graphs $G$ of order at least 3 such that there is a $k$-connected monochromatic subgraph of order at least $n-f(G,k,m)$ in any rainbow $G$-free coloring of $K_n$ using all the $m$ colors. In this paper, we prove that the set $\mathcal{A}$ consists of precisely $P_6$, $P_3\cup P_4$, $K_2\cup P_5$, $K_2\cup 2P_3$, $2K_2\cup K_3$, $2K_2\cup P^{+}_4$, $3K_2\cup K_{1,3}$ and their subgraphs of order at least 3. Moreover, we show that for any graph $H\in \mathcal{A}$, if $n$ sufficiently larger than $m$ and $k…
For integers $n\ge 0$, an iterated triangulation $Tr(n)$ is defined recursively as follows: $Tr(0)$ is the plane triangulation on three vertices and, for $n\ge 1$, $Tr(n)$ is the plane triangulation obtained from the plane triangulation $Tr(n-1)$ by, for each inner face $F$ of $Tr(n-1)$, adding inside $F$ a new vertex and three edges joining this new vertex to the three vertices incident with $F$. In this paper, we show that there exists a 2-edge-coloring of $Tr(n)$ such that $Tr(n)$ contains no monochromatic copy of the cycle $C_k$ for any $k\ge 5$. As a consequence, the answer to one of two …
Let the diameter cover number, $D^t_r(G)$, denote the least integer $d$ such that under any $r$-coloring of the edges of the graph $G$, there exists a collection of $t$ monochromatic subgraphs of diameter at most $d$ such that every vertex of $G$ is contained in at least one of the subgraphs. We explore the diameter cover number with two colors and two subgraphs when $G$ is a complete multipartite graph with at least three parts. We determine exactly the value of $D_2^2(G)$ for all complete tripartite graphs $G$, and almost all complete multipartite graphs with more than three parts.
For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to wonder how "large" of a monochromatic copy of $G$ we can find with respect to some measure -- for instance, the density (or upper density) of the vertex set of $G$ in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked. In one of the few resu…
We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertices. Improving an upper bound of Axenovich and Iverson, we show that maxR(n,K_m,K_4) <= n^{3/2}\sqrt{2m} for all m >= 3. Further, we discuss a possible way to improve their lower bound on maxR(n,K_4,K_4) based on incidence graphs of finite projective planes.
For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free $k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$ is monochromatic. We study the complexity of deciding for fixed $F$ and $k$ whether a given simple digraph admits an $F$-free $k$-coloring. Our main focus is on the restriction of the problem to planar input digraphs, where it is only interesting to study the cases $k \in \{2,3\}$. From known results it follows that for every fixed digraph $F$ whose underlying graph is not a forest, every planar digraph $D$ admits an $F$-free $2$-coloring, a…
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…
A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochro…
We show that for every $η>0$, there exists $n_0$ such that for every even $n$, $n\ge n_0$, and every graph $G$ with $(2+η)n$ vertices and minimum degree at least $(7/4+4η)n$, each colouring of the edges of $G$ with three colours results in a monochromatic cycle of length $n$.
A classical result of Erdős, Gyárfás and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.
The solution to the problem of finding the minimum number of monochromatic triples $(x,y,x+ay)$ with $a\geq 2$ being a fixed positive integer over any 2-coloring of $[1,n]$ was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky's proof (2003) on the minimum number of monochromatic Schur triples $(x,y,x+y)$. We do this by exploiting the combinatorial nature of the original proof and adapting it to the general problem.
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.
We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph contains a monochromatic connected subgraph with at least $1/6$ of the edges. We further show that such a coloring has a monochromatic circuit with a fraction $1/6-o(1)$ of the edges. This verifies a conjecture of Conlon and Tyomkyn. Moreover, for general $k$, we show that every $k$-coloring of the edges of $K_n$ contains a monochromatic connected subgraph wi…
We show that, for every $r, k$, there is an $n = n(r,k)$ so that any $r$-coloring of the edges of the complete graph on $[n]$ will yield a monochromatic complete subgraph on vertices ${a + \sum_{i \in I} d_i \mid I \subseteq [k]}$ for some choice of $a, d_1,..., d_k$. In particular, there is always a solution to $x_1 + ... + x_\ell = y_1 + ... + y_\ell$ whose induced subgraph is monochromatic.
Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be partitioned into two vertex-disjoint cycles, one of each colour. We prove that this conjecture holds for $n$ large enough, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.
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.
For every $n\in\mathbb{N}$ and $k\geq2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geq3$, it is known that the complete graph can be replaced by a graph $G$ with $δ(G)\geq(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gyárfas and Sárközy.
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.
We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\{x,y,x+ay\}$ triples for $a \geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum number of such triples is at most $\frac{n^2}{2a(a^2+2a+3)} + O(n)$ when $a \geq 2$. We also find a new upper bound for the minimum number, over all $r$-colorings of $[1,n]$, of monochromatic Schur triples, for $r \geq 3$.
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…
Consider a graph $G$ drawn on a fixed surface, and assign to each vertex a list of colors of size at least two if $G$ is triangle-free and at least three otherwise. We prove that we can give each vertex a color from its list so that each monochromatic connected subgraph has bounded weak diameter (i.e., diameter measured in the metric of the whole graph $G$, not just the subgraph). In case that $G$ has bounded maximum degree, this implies that each connected monochromatic subgraph has bounded size. This solves a problem of Esperet and Joret for planar triangle-free graphs, and extends known res…
Lehel conjectured that in every $2$-coloring of the edges of $K_n$, there is a vertex disjoint red and blue cycle which span $V(K_n)$. Łuczak, Rödl, and Szemerédi proved Lehel's conjecture for large $n$, Allen gave a different proof for large $n$, and finally Bessy and Thomassé gave a proof for all $n$. Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed a significant strengthening of Lehel's conjecture where $K_n$ is replaced by any graph $G$ with $δ(G)> 3n/4$; if true, this minimum degree condition is essentially best possible. We prove that their conjecture holds when $δ(G)>(3/4+o(1))n$. …
Erdős and Pach (1983) introduced the natural degree-based generalisations of Ramsey numbers, where instead of seeking large monochromatic cliques in a $2$-edge coloured complete graph, we seek monochromatic subgraphs of high minimum or average degree. Here we expand the study of these so-called quasi-Ramsey numbers in a few ways, in particular, to multiple colours and to uniform hypergraphs. Quasi-Ramsey numbers are known to exhibit a certain unique phase transition and we show that this is also the case across the settings we consider. Our results depend on a density-biased notion of hypergra…
Given a $k$-colouring of the edges of the complete graph $K_n$, are there $k-1$ monochromatic components that cover its vertices? This important special case of the well-known Lovász-Ryser conjecture is still open. In this paper we consider a strengthening of this question, where we insist that the covering sets are not merely connected but have bounded diameter. In particular, we prove that for any colouring of $E(K_n)$ with 4 colours, there is a choice of sets $A_1, A_2, A_3$ that cover all vertices, and colours $c_1, c_2, c_3$, such that for each $i = 1,2,3$ the monochromatic subgraph induc…
Consider the following random process: The vertices of a binomial random graph $G_{n,p}$ are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number $r$ of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph $F$ in the process. Our first main result is that for any $F$ and $r$, the threshold function for this problem is given by $p_0(F,r,n)=n^{-1/m_1^*(F,r)}$, where $m_1^*(F,r)$ denotes the so-called \emph{online vertex-Ramsey densit…
We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $Δ(F_i)\leq Δ$ hold for all $i$. Then in any $r$-edge coloured complete graph $K_n$, there is a collection of at most $\exp(C_rΔ)$ monochromatic subgraphs, each of which is isomorphic to an element of $\mathcal{F}$, whose vertex sets partition $V(K_n)$. This proves a conjecture of Corsten and Mendonça in a strong form and generalizes results on the multicolour Ramsey numbers of bounded-…
Let G be a simple balanced bipartite graph on $2n$ vertices, $δ= δ(G)/n$, and $ρ={δ+ \sqrt{2 δ-1} \over 2}$. If $δ> 1/2$ then it has a $\lfloor ρn \rfloor$-regular spanning subgraph. The statement is nearly tight.
Let $G$ be an $r$-uniform hypergraph on $n$ vertices such that all but at most $\varepsilon \binom{n}{\ell}$ $\ell$-subsets of vertices have degree at least $p \binom{n-\ell}{r-\ell}$. We show that $G$ contains a large subgraph with high minimum $\ell$-degree.
Let G be a finite simple graph with automorphism group A(G). Then a spanning subgraph U of G is a fixing subgraph of G if G contains exactly $| A(G)|/ | A(G) \cap A(U)| $ subgraphs isomorphic to U: the graph G must always contain at least this number. If in addition $A(U) \subseteq A(G)$ then U is a strong fixing subgraph. Fixing subgraphs are important in many areas of graph theory. We consider them in the context of Hamiltonian graphs
What is the chance that among a group of $n$ friends, there are $s$ friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic $s$-clique $K_s$ ($s$-matching birthdays) in the complete graph $K_n$, where every vertex of $K_n$ is uniformly colored with $365$ colors (corresponding to birthdays). More generally, for a general connected graph $H$, let $T(H, G_n)$ be the number of monochromatic copies of $H$ in a uniformly random coloring of the vertices of the graph $G_n$ with $c_n$ colors. In this paper we show t…
We study graphs on $n$ vertices which have $2n-2$ edges and no proper induced subgraphs of minimum degree $3$. Erdős, Faudree, Gyárfás, and Schelp conjectured that such graphs always have cycles of lengths $3,4,5,\dots, C(n)$ for some function $C(n)$ tending to infinity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with $n$ vertices and $2n-2$ edges, containing no proper subgraph of minimum degree $3$.
The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family F (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph K_n with t colors. Another area is to find the minimum number of monochromatic members of F that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for fixed positive integers s,t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of …
In 1990 Erdős, Faudree, Rousseau and Schelp proved that for $k\geq 2$, every graph with $n\geq k+1$ vertices and $(k-1)(n-k+2)+\binom{k-2}{2}+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n}/\sqrt{6k^3}$ vertices. They conjectured that it is possible to remove at least $ε_k n$ many vertices and remain with a subgraph of minimum degree $k$, for some $ε_k>0$. We make progress towards their conjecture by showing that one can remove at least $Ω(n/\log n)$ many vertices.
Erdős, Faudree, Rousseau and Schelp observed the following fact for every fixed integer $k\geq 2$: Every graph on $n\geq k-1$ vertices with at least $(k-1)(n-k+2)+{k-2\choose 2}$ edges contains a subgraph with minimum degree at least $k$. However, there are examples in which the whole graph is the only such subgraph. Erdős et al. conjectured that having just one more edge implies the existence of a subgraph on at most $(1-\varepsilon_k)n$ vertices with minimum degree at least $k$, where $\varepsilon_k>0$ depends only on $k$. We prove this conjecture, using and extending ideas of Mousset, Noeve…
