Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of the vertex set of $G$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called \emph{rainbow vertex-disconnected} if for any two vertices $x$ and $y$ of $G$, there exists a vertex subset $S$ such that when $x$ and $y$ are nonadjacent, $S$ is rainbow and $x$ and $y$ belong to different components of $G-S$; whereas when $x$ and $y$ are adjacent, $S+x$ or $S+y$ is rainbow and $x$ and $y$ belong to different components of $(G-xy)-S$. Such a vertex subset $S$ is called a \emph{rainbow vertex-cut} of $G$. For a connected graph $G$, the \emph{rainbow vertex-disconnection number} of $G$, denoted by $rvd(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-disconnected.
In this paper, we obtain bounds of the rainbow vertex-disconnection number of a graph in terms of the minimum degree and maximum degree of the graph. We give a tighter upper bound for the maximum size of a graph $G$ with $rvd(G)=k$ for $k\geq\frac{n}{2}$. We then characterize the graphs of order $n$ with rainbow vertex-disconnection number $n-1$ and obtain the maximum size of a graph $G$ with $rvd(G)=n-1$. Moreover, we get a sharp threshold function for the property $rvd(G(n,p))=n$ and prove that almost all graphs $G$ have $rvd(G)=rvd(\overline{G})=n$. Finally, we obtain some Nordhaus-Gaddum-type results: $n-5\leq rvd(G)+rvd(\overline{G})\leq 2n$ and $n-1\leq rvd(G)\cdot rvd(\overline{G})\leq n^2$ for the rainbow vertex-disconnection numbers of nontrivial connected graphs $G$ and $\overline{G}$ with order $n\geq 24$.
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…
Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of the vertex set of $G$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called \emph{rainbow vertex-disconnected} if for any two vertices $x$ and $y$ of $G$, there exists a vertex subset $S$ of $G$ such that when $x$ and $y$ are nonadjacent, $S$ is rainbow and $x$ and $y$ belong to different components of $G-S$; whereas when $x$ and $y$ are adjacent, $S+x$ or $S+y$ is rainbow and $x$ and $y$ belong to different components of $(G-xy)-S$. For a connected graph $G$, the \emph{rainbow vert…
Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph $G$ is rainbow disconnected if for every two vertices $u$ and $v$, there exists a $u-v$ rainbow cut. 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 solve a conjecture that determines the maximum size of a connected graph $G$ of order $n$ with $rd(G) = k$ for given …
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…
Let $G$ be a nontrivial connected, edge-colored graph. An edge-cut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edge-coloring of $G$ is a rainbow disconnection coloring if for every two distinct vertices $s$ and $t$ of $G$, there exists a rainbow cut $S$ in $G$ such that $s$ and $t$ belong to different components of $G\setminus S$. For a connected graph $G$, the {\it rainbow disconnection number} of $G$, denoted by $rd(G)$, is defined as the smallest number of colors such that $G$ has a rainbow disconnection coloring by using this number of colors…
Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a {\it rainbow edge-cut} if no two edges of $R$ are colored with the same color. For two distinct vertices $u$ and $v$ of $G$, if an edge-cut separates them, then the edge-cut is called a {\it $u$-$v$-edge-cut}. An edge-colored graph $G$ is called \emph{strong rainbow disconnected} if for every two distinct vertices $u$ and $v$ of $G$, there exists a both rainbow and minimum $u$-$v$-edge-cut ({\it rainbow minimum $u$-$v$-edge-cut} for short) in $G$, separating them, and this edge-coloring is called a {\it st…
The concept of rainbow disconnection number of graphs was introduced by Chartrand et al. in 2018. Inspired by this concept, we put forward the concepts of rainbow vertex-disconnection and proper disconnection in graphs. In this paper, we first show that it is $NP$-complete to decide whether a given edge-colored graph $G$ with maximum degree $Δ(G)=4$ is proper disconnected. Then, for a graph $G$ with $Δ(G)\leq 3$ we show that $pd(G)\leq 2$ and determine the graphs with $pd(G)=1$ and $2$, respectively. Furthermore, we show that for a general graph $G$, deciding whether $pd(G)=1$ is $NP$-complete…
Let G be a vertex-colored graph. A vertex cut S of G is called a monochromatic vertex cut if the vertices of S are colored with the same color. A graph G is monochromatically vertex-disconnected if any two nonadjacent vertices of G has a monochromatic vertex cut separating them. The monochromatic vertex-disconnection number of G, denoted by mvd(G), is the maximum number of colors that are used to make G monochromatically vertex-disconnected. In this paper, the connection between the graph parameters are studied: mvd(G), connectivity and block decomposition. We determine the value of mvd(G) for…
Let G be a vertex-colored graph. We call a vertex cut S of G a monochromatic vertex cut if the vertices of S are colored with the same color. The graph G is monochromatically vertex-disconnected if any two nonadjacent vertices of G has a monochromatic vertex cut separating them. The monochromatic vertex-disconnection number of G, denoted by mvd(G), is the maximum number of colors that are used to make G monochromatically vertex-disconnected. In this paper, we propose an algorithm to compute mvd(G) and give an mvd-coloring of the graph G. We run this algorithm with an example written in Java. T…
A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path connecting them. For a connected graph $G$, the \emph{rainbow vertex-connection number} of $G$, denoted by $rvc(G)$, is defined as the minimum number of colors that are required to make $G$ rainbow vertex-connected. In this paper, we find all the families $\mathcal{F}$ of connected graphs with $|\mathcal{F}|\in\{1,2\}$, for which there is a constant $k_\math…
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…
A vertex-colored graph $G$ is rainbow vertex-connected if any pair of distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of $G$, denoted by $rvc(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-connected. In this paper we give a Nordhaus-Gaddum-type result of the rainbow vertex-connection number. We prove that when $G$ and $\bar{G}$ are both connected, then $2\leq rvc(G)+rvc(\bar{G})\leq n-1$. Examples are given to show that both the upper bound and the lower bound are best possible for all…
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 $M=K_2$.
A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strong rainbow vertex-connected. The minimum number $k$ for which there exists a $k$-vertex-coloring of $G$ that result…
The $k$-rainbow index $rx_k(G)$ of a connected graph $G$ was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the $k$-rainbow index, we introduced the concept of $k$-vertex-rainbow index $rvx_k(G)$ in this paper. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a \emph{vertex-rainbow $S$-tree}…
The {\em rainbow vertex-connection number}, $rvc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its vertices such that every pair of vertices is connected by at least one path whose internal vertices have distinct colors. In this paper we first determine the rainbow vertex-connection number of cycle $C_n$ of order $n\geq 3$, and then, based on it, prove that for any 2-connected graph $G$, $rvc(G)\leq rvc(C_n)$, giving a tight upper bound for the rainbow vertex-connection. As a consequence, we show that for a connected graph $G$ with a block decomposition $B_1, B_…
For a given graph $H$ we define $ρ(H)$ to be the minimum order of a graph $G$ such that every proper vertex coloring of $G$ contains a rainbow induced subgraph isomorphic to $H$. We give upper and lower bounds for $ρ(H)$, compute the exact value for some classes of graphs, and consider an interesting combinatorial problem connected with computation of $ρ(H)$ for paths. This research is motivated by some ideas in on-line graph coloring algorithms.
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.
For a positive integer $k$, a $k$-rainbow dominating function ($k$RDF) on a digraph $D$ is a function $f$ from the vertex set $V(D)$ to the set of all subsets of $\{1,2,\ldots,k\}$ such that for any vertex $v$ with $f(v)=\emptyset$, $\bigcup_{u\in N^-(v)}f(u)=\{1,2,\ldots,k\}$, where $N^-(v)$ is the set of in-neighbors of $v$. The weight of a $k$RDF $f$ is defined as $\sum_{v\in V(D)}|f(v)|$. A $k$RDF $f$ on $D$ with no isolated vertex is called a total $k$-rainbow dominating function if the subdigraph of $D$ induced by the set $\{v\in V(D):f(v)\ne\emptyset\}$ has no isolated vertex. The total…
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 vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. Krivelevich and Yuster proved that if $G$ is a graph of order $n$ with minimum degree $δ$, then $rvc(G)<11n/δ$. In this paper, we show that $rvc(G)\leq 3n/(δ+1)+5$ for $δ\geq \sqrt{n-1}-1$ and $n\geq 290$, while $rvc(G)\leq 4n…
A rainbow graph is a graph that admits a vertex-coloring such that every color appears exactly once in the neighborhood of each vertex. We investigate some properties of rainbow graphs. In particular, we show that there is a bijection between the isomorphism classes of n-rainbow graphs on 2n vertices and the switching classes of graphs on n vertices.
A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough (namely, $n\geq 4.25k^2$), then each $n$-vertex graph $G$ with minimum color degree at least $k$ contains a rainbow matching of size at least $k$.
Let $G$ be a connected graph of chromatic number $k$. For a $k$-coloring $f$ of $G$, a full $f$-rainbow path is a path of order $k$ in $G$ whose vertices are all colored differently by $f$. We show that $G$ has a $k$-coloring $f$ such that every vertex of $G$ lies on a full $f$-rainbow path, which provides a positive answer to a question posed by Lin (Simple proofs of results on paths representing all colors in proper vertex-colorings, Graphs Combin. 23 (2007) 201-203). Furthermore, we show that if $G$ has a cycle of length $k$, then $G$ has a $k$-coloring $f$ such that, for every vertex $u$ o…
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 $G = (V, E)$ 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 establish a corresponding result for fixed even rainbow $\ell$-cycles $C_{\ell}$: if every vertex $v \in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n \geq n_0(\ell)$ is sufficiently large, then $G$ admits an even rainbow $\ell$-cycle $C_{\ell}$. This result is best possible whenever $\ell \not\equiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even o…
For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\rightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors. In other words, if $G\rightarrow H$ then a totally multicolored induced copy of $H$ is unavoidable in any vertex-coloring of $G$ with $k$ colors. In this paper, we show that, with a few notable exceptions, for any graph $H$ on $k$ vertices and for any graph $G$ which is not isomorphic to $H$, $G\not\!\rightarrow H$. We explicitly describe all exceptional ca…
A vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In this paper, we study the computational complexity of vertex-rainbow connection of graphs and prove that computing $rvc(G)$ is NP-Hard. Moreover, we show that it is already NP-Complete to decide whether $rvc(G)=2$. We also p…
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…
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.
For a function $f : V(G ) \rightarrow \{0, 1, 2\}$ we denote by $V_i$ the set of vertices to which the value $i$ is assigned by $f$, i.e. $V_i = \{ x \in V (G ) : f(x ) = i \}$. If a function $f: V(G) \rightarrow \{0,1,2\}$ satisfying the condition that $V_i$ is independent for $i \in \{1,2\}$ and every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = i$ for each $i \in \{1,2\}$, then $f$ is called a 2-rainbow independent dominating function (2RiDF). The weight $w(f)$ of a 2RiDF $f$ is the value $w(f) = |V_1|+|V_2|$. The minimum weight of a 2RiDF on a gr…
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.
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) …
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…
A path in an edge (vertex)-colored graph $G$, where adjacent edges (vertices) may have the same color, is called a rainbow path if no pair of edges (internal vertices) of the path are colored the same. The rainbow (vertex) connection number $rc(G)$ ($rvc(G)$) of $G$ is the minimum integer $i$ for which there exists an $i$-edge (vertex)-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. Denote by $\mathcal{G}_d(n)$ ($\mathcal{G'}_d(n)$) the set of all graphs of order $n$ with rainbow (vertex) connection number $d$, and define $e_d(n)=\min\{e(G)\,|\, G\…
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…
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）
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…
Rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colours needed to colour its edges, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we investigate the relationship of rainbow connection number with vertex and edge connectivity. It is already known that for a connected graph with minimum degree $δ$, the rainbow connection number is upper bounded by $3n/(δ+ 1) + 3$ [Chandran et al., 2010]. This directly gives an upper bound of $3n/(λ+ 1) + 3$ and $3n/(κ+ 1) + 3$ for rainbow connectio…
For a graph $G$, we define $σ_2(G)=min \{d(u)+d(v)| u,v\in V(G), uv\not\in E(G)\}$, or simply denoted by $σ_2$. A edge-colored graph is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors, which was introduced by Chartrand et al. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow edge-connected. We prove that if $G$ is a connected graph of order $n$, then $rc(G)\leq 6\frac{n-2}{σ_2+2}+7$. Moreover, the bound is seen to be tight up to additive factors b…
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. …
We prove that for all nonnegative integers k,s there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gyarfas and Sarkozy, who proved the same (when k=2) for graphs G with girth at least five.
A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection number} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In a previous paper we showed that it is NP-Complete to decide whether a given graph $G$ has $rvc(G)=2$. In this paper we show that for every integer $k\geq 2$, deciding whether $rvc(G)\leq k$ is NP-Hard.…
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 $k$ and $n$ be two integers, with $k\geq 3$, $n\equiv 0\pmod k$, and $n$ sufficiently large. We determine the $(k-1)$-degree threshold for the existence of a rainbow perfect matchings in $n$-vertex $k$-uniform hypergraph. This implies the result of Rödl, Ruciński, and Szemerédi on the $(k-1)$-degree threshold for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs. In our proof, we identify the extremal configurations of closeness, and consider whether or not the hypergraph is close to the extremal configuration. In addition, we also develop a novel absorbing device an…
An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. It was proved that computing $rc(G)$ is an NP-Hard problem, as well as that even deciding whether a graph has $rc(G)=2$ is NP-Complete. It is known that deciding whether a given edge-colored graph is rainbow connected is NP-Complete. We will prove that it is still NP-Complete even when the edge-colored graph …
Let $G_1,...,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $δ(G_i)\ge n/2$. A recent conjecture of Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. a cycle with edge set $\{e_1,...,e_n\}$ such that $e_i\in E(G_i)$ for $1\leq i \leq n$. This can be viewed as a rainbow version of the well-known Dirac theorem. In this paper, we prove this conjecture asymptotically by showing that for every $\varepsilon>0$, there exists an integer $N>0$, such that when $n>N$ for any graphs $G_1,...,G_n$ on the same vertex set of size $n$ with $δ(G_i)\ge (\frac{…
Let k be a positive integer and let f be a map from V(G) to the set of all subsets of {1,2,3,...,k}. The function f is called a k-rainbow dominating function of G provided that whenever u is a vertex of G such that f(u) is the empty set, then for each integer r in {1,2,3,...,k} there is a neighbor x of u such that f(x) contains r. The k-rainbow domination number of G is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by a k-rainbow dominating function of G. The k-rainbow domination number of G is the ordinary domination number of the Cartesian product …
Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most $k$ colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for …
The rainbow Turán number, a natural extension of the well studied traditional Turán number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraëte. The rainbow Turán number of a graph $H$, $ex^{*}(n,H)$, is the largest number of edges for an $n$ vertex graph $G$ which can be properly edge colored with no rainbow $H$ subgraph. We explore the reduction method for finding upper bounds on rainbow Turán numbers, and use this to inform results for the rainbow Turán numbers of double stars, caterpillars, and perfect binary trees. In addition, we define $k$-unique colorings and the related …
Given a family $\mathcal G$ of graphs on a common vertex set $X$, we say that $\mathcal G$ is rainbow connected if for every vertex pair $u,v \in X$, there exists a path from $u$ to $v$ that uses at most one edge from each graph in $\mathcal G$. We consider the case that $\mathcal G$ contains $s$ graphs, each sampled randomly from $G(n,p)$, with $n = |X|$ and $p = \frac{c \log n}{sn}$, where $c > 1$ is a constant. We show that when $s$ is sufficiently large, $\mathcal G$ is a.a.s. rainbow connected, and when $s$ is sufficiently small, $\mathcal G$ is a.a.s. not rainbow connected. We also calcu…
