An edge-coloured cycle is $rainbow$ if all edges of the cycle have distinct colours. For $k\geq 1$, let $\mathcal{F}_{k}$ denote the family of all graphs with the property that any $k$ vertices lie on a cycle. For $G\in \mathcal{F}_{k}$, a $k$-$rainbow$ $cycle$ $colouring$ of $G$ is an edge-colouring such that any $k$ vertices of $G$ lie on a rainbow cycle in $G$. The $k$-$rainbow$ $cycle$ $index$ of $G$, denoted by $crx_{k}(G)$, is the minimum number of colours needed in a $k$-rainbow cycle colouring of $G$. In this paper, we restrict our attention to the computational aspects of $k$-rainbow cycle colouring. First, we prove that the problem of deciding whether $crx_1=3$ can be solved in polynomial time, but that of deciding whether $crx_1 \leq k$ is NP-Complete, where $k\geq 4$. Then we show that the problem of deciding whether $crx_2=3$ can be solved in polynomial time, but those of deciding whether $crx_2 \leq 4$ or $5$ are NP-Complete. Furthermore, we also consider the cases of $crx_3=3$ and $crx_3 \leq 4$. Finally, We prove that the problem of deciding whether a given edge-colouring (with an unbounded number of colours) of a graph is a $k$-rainbow cycle colouring, is NP-Complete for $k=1$, $2$ and $3$, respectively. Some open problems for further study are mentioned.
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph $H$ in a quasirandom host graph $G$, assuming that the edge-colouring of $G$ fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings, for example to graph decompositions, orthogonal double covers and graph …
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.
We study rainbow-free colourings of $k$-uniform hypergraphs; that is, colourings that use $k$ colours but with the property that no hyperedge attains all colours. We show that $p^*=(k-1)(\ln n)/n$ is the threshold function for the existence of a rainbow-free colouring in a random $k$-uniform hypergraph.
One of the most interesting new developments in hypergraph colourings in these last few years has been Voloshin's notion of colourings of mixed hypergraphs. In this paper we shall study a specific instance of Voloshin's idea: a non-monochromatic non-rainbow (NMNR) colouring of a hypergraph is a colouring of its vertices such that every edge has at least two vertices coloured with different colours (non-monochromatic) and no edge has all of its vertices coloured with distinct colours (non-rainbow). Perhaps the most intriguing phenomenon of such colourings is that a hypergraph can have gaps in i…
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…
A subgraph $H$ of an edge-coloured graph is called rainbow if all of the edges of $H$ have different colours. In 1989, Andersen conjectured that every proper edge-colouring of $K_{n}$ admits a rainbow path of length $n-2$. We show that almost all optimal edge-colourings of $K_{n}$ admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture holds for almost all optimal edge-colourings of $K_{n}$ and answers a recent question of Ferber, Jain, and Sudakov. Our result also has applications to the existence of transve…
We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, i.e., a path where no two edges have the same colour. The minimum number of colours required for a rainbow colouring of the edges of G is called the rainbow connection number (or rainbow connectivity) rc(G) of G. We investigate sharp thresholds in the Erdős-Rényi random graph for the property "rc(G) <= r" where r is a fixed integer. It is known that for r=2, rainbow connection number 2 and diameter 2 happen essentially at the same time in random graphs. For r >= 3, we conjecture th…
We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{ω_2}]^2\to 2$ establishing $ω_2 \not\to [(ω_1;ω)]^2_2$ such that some colouring $g:[ω_1]^2\to 2$ can not be embedded into $c$. It is also consistent that $2^{ω_1}$ is arbitrarily large, and a function $g$ establishes $2^{ω_1} \not\to [(ω_1,ω_2)]^2_{ω_1}$ such that there is no uncountable $g$-rainbow subset of $2^{ω_1}$. We also show that for each $k\in ω$ it is consistent with Martin's Axiom that the negative partition relation $ω_1 \not\to^* [(ω_1;ω_1)]_{k-bdd}$ holds.
An edge-coloured path is \emph{rainbow} if all the edges have distinct colours. For a connected graph $G$, the \emph{rainbow connection number} $rc(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow path. Similarly, the \emph{strong rainbow connection number} $src(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow geodesic (i.e., a path of shortest length). These two concepts of connectivity in graphs were introduced by Chartrand et al.~in 2008. Subsequently…
We prove a rainbow version of the blow-up lemma of Komlós, Sárközy and Szemerédi for $μn$-bounded edge colourings. This enables the systematic study of rainbow embeddings of bounded degree spanning subgraphs. As one application, we show how our blow-up lemma can be used to transfer the bandwidth theorem of Böttcher, Schacht and Taraz to the rainbow setting. It can also be employed as a tool beyond the setting of $μn$-bounded edge colourings. Kim, Kühn, Kupavskii and Osthus exploit this to prove several rainbow decomposition results. Our proof methods include the strategy of an alternative proo…
A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. In this article, we show the following: 1. The problem of deciding whether a graph can be rainbow coloured using 3 colours remains NP-complete even when restricted to the class of split graphs. However, any spl…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition. An edge-colouring is locally $k$-bounded if each vertex is contained in at most $k$ edges of the same colour. In this paper we prove that any such edge-colouring of the complete graph $K_n$ contains a rainbow copy of every tree with at most $(1-o…
A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a rainbow path. The minimum $k$ for which there exists such an edge-colouring is the rainbow connection number $rc(G)$ of $G.$ Recently, Bau et al. \cite{BJJKM2018} introduced this concept with the additional requirement that the edge-colouring must be proper. %An proper edge-coloured graph is said to be \emph{properly rainbow connected} if any two distinct vertices…
A famous theorem of Dirac states that any graph on $n$ vertices with minimum degree at least $n/2$ has a Hamilton cycle. Such graphs are called Dirac graphs. Strengthening this result, we show the existence of rainbow Hamilton cycles in $μn$-bounded colourings of Dirac graphs for sufficiently small $μ>0$.
For any $r$-graph $H$, we consider the problem of finding a rainbow $H$-factor in an $r$-graph $G$ with large minimum $\ell$-degree and an edge-colouring that is suitably bounded. We show that the asymptotic degree threshold is the same as that for finding an $H$-factor.
A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one rainbow path. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Between them, Chakraborty et al. [J. Comb. Optim., 2011] and Ananth et al. [FSTTCS, 2012] have shown that for every integer k, k \geq 2, it is NP-complete to decide whether a given graph can be rainbow coloured using k colours. A split graph is a …
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph $K_{2n}$, there exists a decomposition of $K_{2n}$ into isomorphic rainbow spanning trees. This settles conjectures of Brualdi--Hollingsworth (from 1996) and Constantine (from 2002) for large graphs.
The sharp threshold for rainbow stackings of random edge-colourings
Institute for Basic Science（基础科学研究院）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research institute of Intelligent Complex Systems, Fudan University（复旦大学复杂系统智能研究所）；Department of Mathematics, Louisiana State University（路易斯安那州立大学数学系）
We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform colouring of $\mathbb{G}(n,ω(1)/n)$, using a palette of size $n$, a.a.s. admits a rainbow copy of any given bounded-degree tree on at most $(1-\varepsilon)n$ vertices, where $\varepsilon > 0$ is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov pertaini…
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.
An edge-coloured graph $G$ is rainbow connected if there exists a rainbow path between any two vertices. A graph $G$ is said to be $k$-rainbow connected if there exists an edge-colouring of $G$ with at most $k$ colours that is rainbow connected. For integers $n$ and $k$, let $t(n,k)$ denote the minimum number of edges in $k$-rainbow connected graphs of order $n$. In this note, we prove that $t(n,k) = \lceil k(n-2)/(k-1) \rceil$ for all $n, k \ge 3$
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly $n$-edge-coloured complete bipartite graphs $K_{n,n}$ which can be decomposed into rainbow perfect matchings. While there are proper edge-colourings of $K_{n,n}$ without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constra…
Aharoni and Berger conjectured that in every proper edge-colouring of a bipartite multigraph by $n$ colours with at least $n+1$ edges of each colour there is a rainbow matching using every colour. This conjecture generalizes a longstanding problem of Brualdi and Stein about transversals in Latin squares. Here an approximate version of the Aharoni-Berger Conjecture is proved---it is shown that if there are at least $n+o(n)$ edges of each colour in a proper $n$-edge-colouring of a bipartite multigraph then there is a rainbow matching using every colour.
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine's Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of $K_n$ there are $10^{-6}n$ edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conje…
An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. Krivelevich and Yuster generalised this concept to the vertex-coloured setting. Similarly, Liu, Mestre and Sousa introduced the version which involves total-colourings. Dorbec,…
In a graph $G$ with a given edge colouring, a rainbow path is a path all of whose edges have distinct colours. The minimum number of colours required to colour the edges of $G$ so that every pair of vertices is joined by at least one rainbow path is called the rainbow connection number $rc(G)$ of the graph $G$. For any graph $G$, $rc(G) \ge diam(G)$. We will show that for the Erdős-Rényi random graph $G(n,p)$ close to the diameter 2 threshold, with high probability if $diam(G)=2$ then $rc(G)=2$. In fact, further strengthening this result, we will show that in the random graph process, with hig…
An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph $H$, the rainbow Turán number $\mathrm{ex}^{\ast}(n,H)$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $H$. We study the rainbow Turán number of even cycles, and prove that for every fixed $\varepsilon > 0$, there is a constant $C(\varepsilon)$ such that every properly edge-colored graph on $n$ vertices with at least $C(\varepsilon) n^{1 + \varepsilon}$ edges contains a rainbow cycle of even length at most $2 \lceil \frac{\ln …
We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n+1}$, without any rainbow cycle.
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.
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then. In this paper we consider two related questions concerning rainbow subgraphs of complete, edge-coloured graphs and digraphs. In the first part, we show that every properly edge-coloured complete directed graph contains a directed rainbow cycle of length $n-O(n^{4/5})$. This is motivated by an old problem of Hahn and improves a result of Gyarfas and Sarkozy.…
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 all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $δ^c(G)>\frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $δ^c(G)>\frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $n\geq 8k-18$ and $δ^c(G)>\frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of len…
We prove that every properly edge-colored $n$-vertex graph with average degree at least $100(\log n)^2$ contains a rainbow cycle, improving upon $(\log n)^{2+o(1)}$ bound due to Tomon. We also prove that every properly colored $n$-vertex graph with at least $10^5 k^2 n^{1+1/k}$ edges contains a rainbow $2k$-cycle, which improves the previous bound $2^{ck^2}n^{1+1/k}$ obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős--Simonovits supersaturation theorem for even cycles, which may be of independent inte…
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost $n$. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured $K_n$ has a rainbow cycle of length $n-O(n^{3/4})$. One of the main ingredients of our proof, which is of independent int…
We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph $G$ on $n$ vertices has a rainbow cycle on at least $n - O(n^{3/4})$ vertices, by showing that $G$ has a rainbow cycle on at least $n - O(\log n \sqrt{n})$ vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton …
We prove several results regarding edge-colored complete graphs and rainbow cycles, cycles with no color appearing on more than one edge. We settle a question posed by Ball, Pultr, and Vojtěchovský by showing that if such a coloring does not contain a rainbow cycle of length $n$, where $n$ is odd, then it also does not contain a rainbow cycle of length $m$ for all $m$ greater than $2n^2$. In addition, we present two examples which demonstrate that this result does not hold for even $n$. Finally, we state several open problems in the area.
Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis ensures a rainbow $\ell$-cycle $C_{\ell}$ whenever $n \ge 432 \ell$. This result is sharp for all odd integers $\ell \geq 3$, and extends earlier work of the authors for when $\ell$ is even.
Let $G$ be an edge-colored graph with $n$ vertices. A subgraph $H$ of $G$ is called a rainbow subgraph of $G$ if the colors of each pair of the edges in $E(H)$ are distinct. We define the minimum color degree of $G$ to be the smallest number of the colors of the edges that are incident to a vertex $v$, for all $v\in V(G)$. Suppose that $G$ contains no rainbow-cycle subgraph of length four. We show that if the minimum color degree of $G$ is at least $\frac{n+3k-2}{2}$, then $G$ contains a rainbow-cycle subgraph of length at least $k$, where $k\geq 5$. Moreover, if the condition of $G$ is restri…
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…
An edge-colored graph $F$ is {\it rainbow} if each edge of $F$ has a unique color. The {\it rainbow Turán number} $\mathrm{ex}^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Turán numbers was introduced by Keevash, Mubayi, Sudakov, and Verstraëte. Johnson and Rombach introduced the following rainbow-version of generalized Turán problems: for fixed graphs $H$ and $F$, let $\mathrm{ex}^*(n,H,F)$ denote the maximum number of rainbow copies of $H$ in an $n$-vertex properly edge-colored gra…
Given graphs $F$ and $H$, the generalized rainbow Turán number $\text{ex}(n,F,\text{rainbow-}H)$ is the maximum number of copies of $F$ in an $n$-vertex graph with a proper edge-coloring that contains no rainbow copy of $H$. B. Janzer determined the order of magnitude of $\text{ex}(n,C_s,\text{rainbow-}C_t)$ for all $s\geq 4$ and $t\geq 3$, and a recent result of O. Janzer implied that $\text{ex}(n,C_3,\text{rainbow-}C_{2k})=O(n^{1+1/k})$. We prove the corresponding upper bound for the remaining cases, showing that $\text{ex}(n,C_3,\text{rainbow-}C_{2k+1})=O(n^{1+1/k})$. This matches the known…
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）
Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number of copies of $H$ that a properly edge-coloured graph on $n$ vertices can contain if it has no rainbow subgraph isomorphic to $F$. We determine the order of magnitude of $\operatorname{ex}(n,C_s,$rainbow-$C_t)$ for all $s,t$ with $s\not =3$. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer by showing that $\operatorname{ex}(n,C_{2k},$rainbow-$C_{2k})$ is $Θ(n^{k-1})$ if $k\geq 3$ and $Θ(n^2)$ if…
An edge-coloring of a complete graph with a set of colors $C$ is called completely balanced if any vertex is incident to the same number of edges of each color from $C$. Erdős and Tuza asked in $1993$ whether for any graph $F$ on $\ell$ edges and any completely balanced coloring of any sufficiently large complete graph using $\ell$ colors contains a rainbow copy of $F$. This question was restated by Erdős in his list of ``Some of my favourite problems on cycles and colourings''. We answer this question in the negative for most cliques $F=K_q$ by giving explicit constructions of respective comp…
We consider distance colourings in graphs of maximum degree at most $d$ and how excluding one fixed cycle length $\ell$ affects the number of colours required as $d\to\infty$. For vertex-colouring and $t\ge 1$, if any two distinct vertices connected by a path of at most $t$ edges are required to be coloured differently, then a reduction by a logarithmic (in $d$) factor against the trivial bound $O(d^t)$ can be obtained by excluding an odd cycle length $\ell \ge 3t$ if $t$ is odd or by excluding an even cycle length $\ell \ge 2t+2$. For edge-colouring and $t\ge 2$, if any two distinct edges con…
