A coloured graph is k-ultrahomogeneous if every isomorphism between two induced subgraphs of order at most k extends to an automorphism. A coloured graph is t-tuple regular if the number of vertices adjacent to every vertex in a set S of order at most k depends only on the isomorphism type of the subgraph induced by S.
We classify the finite vertex-coloured k-ultrahomogeneous graphs and the finite vertex-coloured l-tuple regular graphs for k at least 4 and l at least 5, respectively. Our theorem in particular classifies finite vertex-coloured ultrahomogeneous graphs, where ultrahomogeneous means the graph is simultaneously k-ultrahomogeneous for all k.
We (re-)prove that in every 3-edge-coloured tournament in which no vertex is incident with all colours there is either a cyclic rainbow triangle or a vertex dominating every other vertex monochromatically.
Bousquet, Lochet and Thomassé recently gave an elegant proof that for any integer $n$, there is a least integer $f(n)$ such that any tournament whose arcs are coloured with $n$ colours contains a subset of vertices $S$ of size $f(n)$ with the property that any vertex not in $S$ admits a monochromatic path to some vertex of $S$. In this note we provide a lower bound on the value $f(n)$.
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.
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…
Let $G$ be an edge-coloured graph. The minimum colour degree $δ^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly coloured if no two adjacent edges have the same colour. In this paper, we show that, for any $\varepsilon >0$ and $n$ large, every edge-coloured graph $G$ with $δ^c(G) \ge (1/2+\varepsilon)n$ contains a properly coloured cycle of length at least $\min\{ n , \lfloor 2 δ^c(G)/3 \rfloor\}$.
An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number $r$, we show that the vertex set of any $r$-locally coloured complete graph may be partitioned into $O(r^2 \log r)$ disjoint monochromatic cycles. This generalises a result of Erdős, Gyárfás and Pyber.
The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and Sivakoff, was inspired by a group of people collectively solving a puzzle. It can also be seen as a measure of whether two graphs on a common vertex set are "jointly connected". In this paper we consider the natural generalisation of this process to an arbitrary number of graphs on the same vertex set. We prove that if these graphs are random, then the jigsaw percolation process exhibits a phase transition in terms of the product of the edge probabilities. This generalises a result of Bollobás, Riordan, Slivken and Smi…
In the context of the coloured stochastic vertex model in a quadrant, we identify a family of observables whose averages are given by explicit contour integrals. The observables are certain linear combinations of $q$-moments of the coloured height functions of the model. In a polymer limit, this yields integral representations for moments of partition functions of strict-weak, semi-discrete Brownian, and continuum Brownian polymers with varying beginning and ending points of the polymers.
A 2-edge-coloured graph $G$ is {\bf supereulerian} if $G$ contains a spanning closed trail in which the edges alternate in colours. An {\bf eulerian factor} of a 2-edge-coloured graph is a collection of vertex disjoint induced subgraphs which cover all the vertices of $G$ such that each of these subgraphs is supereulerian. We give a polynomial algorithm to test if a 2-edge-coloured graph has an eulerian factor and to produce one when it exists. A 2-edge-coloured graph is {\bf (trail-)colour-connected} if it contains a pair of alternating $(u,v)$-paths ($(u,v)$-trails) whose union is an alterna…
Using the definition of colouring of $2$-edge-coloured graphs derived from $2$-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$-edge-coloured graphs. We find closed forms for the first three coefficients of this polynomial that generalize the known results for the chromatic polynomial of a graph. We classify those $2$-edge-coloured graphs that have a chromatic polynomial equal to the chromatic polynomial of the underlying graph, when every vertex is incident to edges of both colours. Finally, we examine the behaviour of the roots of this polynomial, hig…
Let $G$ be a graph whose edges are each assigned one of the $m$-colours $1, 2, \ldots, m$, and let $Γ$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ with respect $π\in Γ$ permutes the colours of the edges incident with $x$ according to $π$. There is a well-developed theory of switching when $Γ$ is Abelian. Much less is known for non-Abelian groups. In this paper we consider switching with respect to non-Abelian groups including symmetric, alternating and dihedral groups. We first consider the question of whether there is a sequence of switches using elements of $Γ$ that tr…
A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on $n$ vertices with minimum colour degree at least $k$ contains a rainbow matching of size at least $k$, provided $n\geq (17/4)k^2$. In this paper, we show that $n\geq 4k-4$ is sufficient for $k \ge 4$.
Let $K_n^c$ be an edge-coloured complete graph on $n$ vertices. Let $Δ_{\rm mon}(K_n^c)$ denote the largest number of edges of the same colour incident with a vertex of $K_n^c$. A properly coloured cycle is a cycle such that no two adjacent edges have the same colour. In 1976, Bollobás and Erdős conjectured that every $K_n^c$ with $Δ_{\rm mon}(K_n^c) < \lfloor n/2 \rfloor$ contains a properly coloured Hamiltonian cycle. In this paper, we show that for any $\varepsilon > 0 $, there exists an integer $n_0$ such that every $K_n^c$ with $Δ_{\rm mon}(K_n^c) < (1/2 - \varepsilon) n $ and $n \ge n_0$…
We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of $G$ contains at most $n/2$ vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but $o(n)$ vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded.…
We prove that for all integers $Δ,r \geq 2$, there is a constant $C = C(Δ,r) >0$ such that the following is true for every sequence $\mathcal{F} = \{F_1, F_2, \ldots\}$ of graphs with $v(F_n) = n$ and $Δ(F_n) \leq Δ$, for each $n \in \mathbb{N}$. In every $r$-edge-coloured $K_n$, there is a collection of at most $C$ monochromatic copies from $\mathcal{F}$ whose vertex-sets partition $V(K_n)$. This makes progress on a conjecture of Grinshpun and Sárközy.
Let $G$ be an edge-coloured graph. The minimum colour degree $ δ^c(G) $ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly coloured if no two adjacent edges have the same colour. In this paper, we show that every edge-coloured graph $G$ with $ δ^c(G) \ge 2|G| / 3 $ contains a properly coloured $2$-factor. Furthermore, we show that for any $ \varepsilon > 0 $ there exists an integer $ n_0 $ such that every edge-coloured graph $G$ with $|G| = n \ge n_0 $ and $ δ^c(G) \ge ( 2/3 + \varepsi…
A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r > 2. In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gyárfás. As an intermediate result we s…
The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set of coloured vertices that can eventually force the entire graph to be coloured. The zero forcing number is the size of the smallest zero forcing set. We explore the zero forcing number for random regular graphs, improving on bounds given by Kalinowski, Kamucev and Sudakov. We also propose and analyze a degree-greedy algorithm for finding small zero forcing set…
This work is dedicated to $\mathfrak{sl}_{n+1}$-related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following: (1) We construct the basis of (rational) eigenfunctions of the coloured transfer-matrices as partition functions of our lattice models with certain boundary conditions. Similarly, we construct a dual basis and prove the corresponding orthogonality relations and Plancherel formulae; (2) We derive a variety of combinatorial properties of those eigenfunctions, such as branching rules, exchange relations…
Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an $m$-coloured complete subgraph for every natural number $m$ or there exists an infinite subset $X \subset \mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on $X$ or there exists a distinguished vertex $v$ in $X$ such that $X \setminus \lbrace v \rbrace$ is $1$-coloured and each edge betwee…
Assume $L$ is a $k$-list assignment of a graph $G$. A $d$-defective $m$-fold $L$-colouring $ϕ$ of $G$ assigns to each vertex $v$ a set $ϕ(v)$ of $m$ colours, so that $ϕ(v) \subseteq L(v)$ for each vertex $v$, and for each colour $i$, the set $\{v: i \in ϕ(v)\}$ induces a subgraph of maximum degree at most $d$. In this paper, we consider on-line list $d$-defective $m$-fold colouring of graphs, where the list assignment $L$ is given on-line, and the colouring is constructed on-line. To be precise, the $d$-defective $(k,m)$-painting game on a graph $G$ is played by two players: Lister and Painter…
An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected} if any two vertices are connected by a rainbow geodesic. The (\emph{strong}) \emph{rainbow connection number} of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. As an extension, Krivelevich and Yuster proposed the…
Given an $n$-vertex graph $G$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $G \cup \mathbb{G}(n,p)$ over the supergraphs of $G$ is referred to as a (random) {\sl perturbation} of $G$. We consider the distribution of edge-coloured graphs arising from assigning each edge of the random perturbation $G \cup \mathbb{G}(n,p)$ a colour, chosen independently and uniformly at random from a set of colours of size $r := r(n)$. We prove that such edge-coloured graph distributions a.a.s. admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisf…
P. Erdős proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R. Rado to an arbitrary finite number of colours. We prove that the vertices of every finite-edge coloured infinite complete graph can be partitioned into disjoint monochromatic paths of different colours. This answers a question of R. Rado from 1978.
Erdős, Gyárfás and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 \log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = Ω(n^{-1/(2r)})$, then with high probability any $r$-edge-coloured $G(n,p)$ can be covered by at most $1000 r^4 \log r $ vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.
A matrix A is image partition regular over Q provided that whenever Q - {0} is finitely coloured, there is a vector x with entries in Q - {0} such that the entries of Ax are monochromatic. It is kernel partition regular over Q provided that whenever Q - {0} is finitely coloured, the matrix has a monochromatic member of its kernel. We establish a duality for these notions valid for both finite and infinite matrices. We also investigate the extent to which this duality holds for matrices partition regular over proper subsemigroups of Q.
A weighted coloured-edge graph is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such a graph paths are compared by their total weight in each colour, resulting in a Pareto set of minimal paths from one vertex to another. This paper will give a tight upper bound on the cardinality of a minimal set of paths for any weighted coloured-edge graph. Additionally, a bound is presented on the expected number of minimal paths in weighted bicoloured-ed…
Universidade Federal de Pernambuco（伯南布哥联邦大学）；Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）；Universidade Federal do Ceará（塞阿拉联邦大学）；Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；Institute for Basic Science（基础科学研究院）
The upper density of an infinite graph $G$ with $V(G) \subseteq \mathbb{N}$ is defined as $\overline{d}(G) = \limsup_{n \rightarrow \infty}{|V(G) \cap \{1,\ldots,n\}|}/{n}$. Let $K_{\mathbb{N}}$ be the infinite complete graph with vertex set $\mathbb{N}$. Corsten, DeBiasio, Lamaison and Lang showed that in every $2$-edge-colouring of $K_{\mathbb{N}}$, there exists a monochromatic path with upper density at least $(12 + \sqrt{8})/17$, which is best possible. In this paper, we extend this result to $k$-edge-colouring of $K_{\mathbb{N}}$ for $k \ge 3$. We conjecture that every $k$-edge-coloured $…
The rainbow Turán number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$, $\mathrm{ex}^*(n,C_{2k})=O(n^{1+1/k})$. This is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstraëte. We use the same method to prove several other conjectures in various topics. First, we prove that there exists a constant $c$ such that any properly edge-coloured $n$-vertex graph with more than $cn(\log n)^4$ edges contains a rainbow cycle. …
Given an $r$-edge-coloured complete graph $K_n$, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an $r$-edge-coloured random graph $\mathcal{G}(n,p)$. Recently, Bucić, Korándi and Sudakov established…
The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colors. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly coloured with the numbers 1,...,4 in such a way that every face contains a unique vertex coloured with the maximal color appearing on that face. They proved that every plane graph has such a colouring with the numbers 1,...,6. We prove that every plane graph has such a colouring with the numbers 1,...,5 and we also prove the list variant of the statement f…
Let ${\cal H}$ denote the family of all graphs with multi-$4$-cycles and suppose that $G \in {\cal H}$. Then, $G$ is a bipartite graph with a vertex bipartition $\{V_α, V_β\}$. We prove that for every vertex $v \in V_β$ and for every $2$-colouring $V_α \rightarrow \{1, 2\}$ there exists a $2$-colouring $V_β \rightarrow \{1, 2\}$ such that every cycle in $G$ is not monochromatic and $b(v) = 1$ ($b(v) = 2$). Let now $G$ be a simple even plane triangulation with a vertex $3$-partition $\{V_{1}, V_{2}, V_{3}\}$. Denote by $B_{i}$, $i = 1, 2, 3$, the set of all vertices in $V_i$ of degree at least …
We prove several results on approximate decompositions of edge-coloured quasirandom graphs into rainbow spanning structures. More precisely, we say that an edge-colouring of a graph is locally $\ell$-bounded if no vertex is incident to more than $\ell$ edges of any given colour, and that it is (globally) $g$-bounded if no colour appears more than $g$ times in the colouring. Note that every proper colouring of an $n$-vertex graph is locally $1$-bounded, and (globally) $n/2$-bounded. Our results imply the following: (i) The existence of approximate decompositions of edge-coloured $K_n$ into rain…
We prove that every $n$-vertex graph with at least $\binom{n}{2} - (n - 4)$ edges has a fractional triangle decomposition, for $n \ge 7$. This is a key ingredient in our proof, given in a companion paper, that every $n$-vertex $2$-coloured complete graph contains $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, which confirms a conjecture of Erdős.
Let G be a graph on n vertices with maximum degree D. We use the Lovász local lemma to show the following two results about colourings c of the edges of the complete graph K_n. If for each vertex v of K_n the colouring c assigns each colour to at most (n-2)/22.4D^2 edges emanating from v, then there is a copy of G in K_n which is properly edge-coloured by c. This improves on a result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4), 409-433, 2003]. On the other hand, if c assigns each colour to at most n/51D^2 edges of K_n, then there is a copy of G in K_n such that each e…
Let $G$ be an edge-coloured graph. A rainbow subgraph in $G$ is a subgraph such that its edges have distinct colours. The minimum colour degree $δ^c(G)$ of $G$ is the smallest number of distinct colours on the edges incident with a vertex of $G$. We show that every edge-coloured graph $G$ on $n\geq 7k/2+2$ vertices with $δ^c(G) \geq k$ contains a rainbow matching of size at least $k$, which improves the previous result for $k \ge 10$. Let $Δ_{\text{mon}}(G)$ be the maximum number of edges of the same colour incident with a vertex of $G$. We also prove that if $t \ge 11$ and $Δ_{\text{mon}}(G) …
A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.
A hypermap is (hypervertex-) bipartite if its hypervertices can be 2-coloured in such a way that ``neighbouring'' hypervertices have different colours. It is bipartite-uniform if within each of the sets of hypervertices of the same colour, hyperedges and hyperfaces, elements have common valencies. The flags of a bipartite hypermap are naturally 2-coloured by assigning the colour of its adjacent hypervertices. A hypermap is bipartite-regular if the automorphism group acts transitively on each set of coloured flags. If the automorphism group acts transitively on the set of all flags, the hyperma…
An edge (vertex) coloured graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph $G$ is the smallest number of colours needed for a rainbow edge (vertex) colouring of $G$. In this paper we propose a very simple approach to studying rainbow connectivity in graphs. Using this idea, we give a unified proof of several known results, as well as some new ones.
Superlinear Lower Bounds for Monochromatic Path Partitions
School of Mathematics, Shandong University（山东大学数学学院）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极值组合与概率研究组）；School of Mathematics, Nanjing University（南京大学数学系）
We study a class of edge-coloured graphs, including the chamber systems of buildings and other geometries such as affine planes, from which we build coherent configurations (also known as non-commutative association schemes). The condition we require is that the graph be endowed with a certain distance function, taking its values in the adjacency algebra (itself generated by the adjacency operators). When all the edges are of the same colour, the condition is equivalent to the graph being distance-regular, so our result is a generalization of the classical fact that distance-regular graphs giv…
In this paper, we study two problems related to planar matchings in random bipartite graphs. First, we colour each edge of the complete bipartite graph $K_{n,n}$ uniformly randomly from amongst ${r}$ colours and show that if ${r}$ grows linearly with ${n}$ then the maximum rainbow matching is a non-trivial fraction of ${r}$ with high probability, i.e. with probability converging to one as ${n \rightarrow \infty}$ Next we consider planar matchings in a dependent setting where each vertex is forced to choose exactly one neighbour from amongst all possible choices. We obtain estimates for the lar…
