Let $G = (V,E)$ be an $n$-vertex graph and let $c: E \to \mathbb{N}$ be a coloring of its edges. Let $d^c(v)$ be the number of distinct colors on the edges at $v \in V$ and let $δ^c(G) = \min_{v \in V} \{ d^{c}(v) \}$. H. Li proved that $δ^c(G) > n/2$ guarantees a rainbow triangle in $G$. We give extensions of Li's result to cliques $K_r$ for $r \ge 4$.
In this paper we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge disjoint rainbow Hamilton cycles. We also ask when in the resultant graph every pair of vertices is connected by a rainbow path.
For an arc-colored digraph $D$, define its {\em kernel by rainbow paths} to be a set $S$ of vertices such that (i) no two vertices of $S$ are connected by a rainbow path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a rainbow path in $D$. In this paper, we show that it is NP-complete to decide whether an arc-colored tournament has a kernel by rainbow paths, where a {\em tournament} is an orientation of a complete graph. In addition, we show that every arc-colored $n$-vertex tournament with all its strongly connected $k$-vertex subtournaments, $3\leq k\leq n$, colored with at least…
A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of different colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(2020)] studied the existence of rainbow matchings in edge-colored graph G with average color degree at least 2k, and proved some sufficient conditions for a rainbow marching of size k in G. The sufficient conditions include that |V(G)|>=12k^2+4k, or G is a properly edge-colored graph with |V(G)|>=8k. In this paper, we show that every edge-colored graph G with |V(G)|>=4k-4…
A spanning tree of a properly edge-colored complete graph, $K_n$, is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if $K_{2m}$ is properly $(2m-1)$-edge-colored, then the edges of $K_{2m}$ can be partitioned into $m$ rainbow spanning trees except when $m=2$. By means of an explicit, constructive approach, in this paper we construct $\lfloor \sqrt{6m+9}/3 \rfloor$ mutually edge-disjoint rainbow spanning trees for any positive value of $m$. Not only are the rainbow trees produced, but also some structure of each rainbow spa…
We show that for any integer $t\geq 2$, every properly edge-coloured graph on $n$ vertices with more than $n^{1+o(1)}$ edges contains a rainbow subdivision of $K_t$. Note that this bound on the number of edges is sharp up to the $o(1)$ error term. This is a rainbow analogue of some classical results on clique subdivisions and extends some results on rainbow Turán numbers. Our method relies on the framework introduced by Sudakov and Tomon[2020] which we adapt to find robust expanders in the coloured setting.
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 …
A {\it rainbow matching} in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f(δ) such that a properly edge-colored graph G with minimum degree δand order at least f(δ) must have a rainbow matching of size δ. We answer this question in the affirmative; f(δ) = 6.5δsuffices. Furthermore, the proof provides a O(δ(G)|V(G)|^2)-time algorithm that generates such a matching.
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.
A hypergraph $H$ is properly colored if for every vertex $v\in V(H)$, all the edges incident to $v$ have distinct colors. In this paper, we show that if $H_{1}$, \cdots, $H_{s}$ are properly-colored $k$-uniform hypergraphs on $n$ vertices, where $n\geq3k^{2}s$, and $e(H_{i})>{{n}\choose {k}}-{{n-s+1}\choose {k}}$, then there exists a rainbow matching of size $s$, containing one edge from each $H_i$. This generalizes some previous results on the Erdős Matching Conjecture.
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 balanced edge-coloring of the complete graph is an edge-coloring such that every vertex is incident to each color the same number of times. In this short note, we present a construction of a balanced edge-coloring with six colors of the complete graph on $n=13^k$ vertices, for every positive integer $k$, with no rainbow $K_4$. This solves a problem by Erdős and Tuza.
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…
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 $T_{n}$ 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^{…
Let $G$ be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let the color degree of a vertex $v$ be the number of different colors that are used on the edges incident to $v$, and denote it to be $d^c(v)$. It was shown that if $d^c(v)\geq k$ for every vertex $v$ of $G$, then $G$ has a rainbow path of length at least $\min\{\lceil\frac{2k+1}{3}\rceil,k-1\}$. In the present paper, we consider the properly edge-colored complete graph $K_n$ only and improve the lower bound of the length of the longest rainbow p…
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 $n$.…
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 …
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…
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 …
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…
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.
Considering a natural generalization of the Ruzsa-Szemerédi problem, we prove that for any fixed positive integers $r,s$ with $r<s$, there are graphs on $n$ vertices containing $n^{r}e^{-O(\sqrt{\log{n}})}=n^{r-o(1)}$ copies of $K_s$ such that any $K_r$ is contained in at most one $K_s$. We also give bounds for the generalized rainbow Turán problem $\operatorname{ex}(n, H,$rainbow-$F)$ when $F$ is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on $n$ vertices with $n^{r-1-o(1)}$ copies of $K_r$ such t…
Instituto de Matemática da Universidade Federal de Alagoas（阿拉戈斯联邦大学数学研究所）；Instituto de Matemática Pura e Aplicada（纯数学与应用数学研究所）；Instituto de Computação da Universidade Federal de Alagoas（阿拉戈斯联邦大学计算机研究所）
Let $G$ be a nontrivial connected and edge-colored graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored with a same color. An edge-colored graph $G$ is called rainbow disconnected if for every two distinct vertices $u$ and $v$ of $G$, there exists a $u-v$ rainbow cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by $rd(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we will study the Erdős-Gallai-type results for $rd(G)$, and completely so…
A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if for any two vertices of $G$ there is a rainbow path connecting them. The rainbow connection number of $G$, denoted by $rc(G)$, is defined as the smallest number of colors by using which there is a coloring such that $G$ is rainbow connected. In this paper, we mainly study the rainbow connection number of the line graph of a graph which contains triangles and get two sharp upper bounds for…
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…
Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges of $G$ and the number of colors appearing on $E(G)$, respectively. For a vertex $v\in V(G)$, the \emph{color neighborhood} of $v$ is defined as the set of colors assigned to the edges incident to $v$. A subgraph of $G$ is \emph{rainbow} if all of its edges are assigned with distinct colors. The well-known Mantel's theorem states that a graph $G$ on $n$ vertices contains a triangle if $e(G)\geq\lfloor\frac{n^2}{4}\rfloor+1$. Rademacher (1941) showed that $G$ contains at least $\lfloor\frac{n}{2}\rfloor$ tri…
Alon and Shikhelman initiated the systematic study of the following generalized Turán problem: for fixed graphs $H$ and $F$ and an integer $n$, what is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph? An edge-colored graph is called rainbow if all its edges have different colors. The rainbow Turán number of $F$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $F$. The study of rainbow Turán problems was initiated by Keevash, Mubayi, Sudakov and Verstraëte. Motivated by the above problems, we study the followi…
We provide an algorithm that verifies the optimal colored Tverberg problem for $10$ points in the plane: Every $10$ points in the plane in color classes of size at most $3$ can be partitioned in $4$ rainbow pieces such that their convex hulls intersect in a common point. This is achieved by translating the problem to $k$-partite graphs and using a new algorithm to verify that those graphs do not have a $k$-clique.
Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K_n such that at each vertex every color appears only constantly many times. In 1979, Shearer conjectured that such a coloring c must contain a properly colored copy of G. We establish this conjecture in a strong form, showing that it holds even for graphs G with O(n^(4/3)) cherries and moreover this bound on the number of cherries is best possible up to a constant factor. We also prove that one can find a rainbow copy of such G in every edge-c…
A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colors that are needed in order to make $G$ rainbow connected. Chartrand et al. obtained that $G$ is a tree if and only if $rc(G)=m$, and it is easy to see that $G$ is not a tree if and only if $rc(G)\leq m-2$, where $m$ is the number of edge of $G$. So there is…
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…
A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for $n$ and $C$ large enough, if $G$ is an edge-colored copy of $K_n$ in which each color class has size at most $n/2$, then $G$ has at least $\lfloor n/(C\log n)\rfloor$ edge-disjoint rainbow spanning trees. Here we strengthen this result by showing that if $G$ is any edge-colored graph with $n$ vertices in which each color appears on at most $δ\cdotλ_1/2$ edges, where $δ\geq C\log n$ for $n$ and $C$ sufficiently large and $λ_1$ is the second-smal…
A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colors that are needed in order to make $G$ rainbow connected. In this paper, we provide a new approach to investigate the rainbow connection number of a graph $G$ according to some constraints to its complement graph $\bar{G}$. We first derive that for a connec…
An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph $G$ is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper, we give a sharp upper bound that $rc(G)\leq\lceil\frac{n}{2}\rceil$ for any 2-connected graph $G$ of order $n$, which improves the results of Caro et al. to best possible.
The $t$-colored rainbow saturation number $rsat_t(n,F)$ is the minimum size of a $t$-edge-colored graph on $n$ vertices that contains no rainbow copy of $F$, but the addition of any missing edge in any color creates such a rainbow copy. Barrus, Ferrara, Vandenbussche and Wenger conjectured that $rsat_t(n,K_s) = Θ(n\log n)$ for every $s\ge 3$ and $t\ge \binom{s}{2}$. In this short note we prove the conjecture in a strong sense, asymptotically determining the rainbow saturation number for triangles. Our lower bound is probabilistic in spirit, the upper bound is based on the Shannon capacity of a…
One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdos-Renyi random graph G_{n,p} is around p ~ (log n + log log n) / n. Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3-uniform hypergraph by connecting 3-uniform hypergraphs to edge-colored graphs. In this work, we consider that setting of edge-colored graphs, and prove a result which achieves the best possible first …
An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there exists a $u$-$v$-rainbow-cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by rd$(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we first give some tight upper bounds for rd$(G)$, and moreover, we completely characterize the graphs which…
The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper, generalizes these problems to edge colored graphs with given total color degree. In particular, we find that if a graph $G$ has total color degree $2mn$ and satisfies some other properties, then $G$ contains a matching of size $m$; These other properties include $G$ being triangle-free, $C_4$-free, properly colored, or large enough.
A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. For any two vertices $u$ and $v$ of $G$, a rainbow $u-v$ geodesic in $G$ is a rainbow $u-v$ path of length $d(u,v)$, where $d(u,v)$ is the distance between $u$ and $v$. The graph $G$ is strongly rainbow connected if there exists a rainbow $u-v$ geodesic for any two vertices $u$ and $v$ in $G$. The strong rainbow connection number of $G$, denoted $src(G)$, is the minimum number of colors that are needed in order to make $G$ strong rainbow connected. In thi…
Let $HP_{n,m,k}$ be drawn uniformly from all $k$-uniform, $k$-partite hypergraphs where each part of the partition is a disjoint copy of $[n]$. We let $HP^{(\k)}_{n,m,k}$ be an edge colored version, where we color each edge randomly from one of $\k$ colors. We show that if $\k=n$ and $m=Kn\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if $n$ is even and $m=Kn\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in $G^{(n)…
An edge-colored graph is said to contain a rainbow-$F$ if it contains $F$ as a subgraph and every edge of $F$ is a distinct color. The problem of maximizing edges among $n$-vertex properly edge-colored graphs not containing a rainbow-$F$, known as the rainbow Turán problem, was initiated by Keevash, Mubayi, Sudakov and Verstraëte. We investigate a variation of this problem with the additional restriction that the graph is planar, and we denote the corresponding extremal number by $\ex_{\p}^*(n,F)$. In particular, we determine $\ex_{\p}^*(n,P_5)$, where $P_5$ denotes the $5$-vertex path.
A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A total-colored graph is \emph{total-rainbow connected} if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph $G$, the \emph{total-rainbow connection number} of $G$, denoted by $trc(G)$, is the minimum number of colors required in a total-coloring of $G$ to make $G$ total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a No…
Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum number of colors that are needed to make $G$ rainbow connected is called the rainbow connection number of $G$, denoted by $rc(G)$. In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if $G$ is a connected graph, then $rc(G)\leq 2α(G)-1$. Two examples $G$ are given to show that the …
A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. The strong rainbow connection number $src(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices $u$ and $v$ of $G$ are connected by a rainbow path of length $d(u,v)$. I…
