Let $G$ be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let $d^c(v)$ denote the color degree and $CN(v)$ denote the color neighborhood of a vertex $v$ of $G$. In a previous paper, we showed that if $d^c(v)\geq k$ (color degree condition) for every vertex $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lceil\frac{k+1}{2}\rceil$, and if $|CN(u)\cup CN(v)|\geq s$ (color neighborhood union condition) for every pair of vertices $u$ and $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lceil\frac{s}{3}\rceil+1$. Later, in another paper we first showed that if $k\leq 7$, $G$ has a heterochromatic path of length at least $k-1$, and then, based on this we use induction on $k$ and showed that if $k\geq 8$, then $G$ has a heterochromatic path of length at least $\lceil\frac{3k}{5}\rceil+1$. In the present paper, by using a simpler approach we further improve the result by showing that if $k\geq 8$, $G$ has a heterochromatic path of length at least $\lceil\frac{2k}{3}\rceil+1$, which confirms a conjecture by Saito. We also improve a previous result by showing that under the color neighborhood union condition, $G$ has a heterochromatic path of length at least $\lfloor\frac{2s+4}{5}\rfloor$.
A {\it heterochromatic tree} is an edge-colored tree in which any two edges have different colors. The {\it heterochromatic tree partition number} of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum positive integer $p$ such that whenever the edges of the graph $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $p$ vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of an $r$-edge-colored complete graph.
Let $G$ be an edge-colored graph. A heterochromatic cycle of $G$ is one in which every two edges have different colors. For a vertex $v\in V(G)$, let $CN(v)$ denote the set of colors which are assigned to the edges incident to $v$. In this note we prove that $G$ contains a heterochromatic cycle of length 4 if $G$ has $n\geq 60$ vertices and $|CN(u)\cup CN(v)|\geq n-1$ for every pair of vertices $u$ and $v$ of $G$. This extends a result of Broersma et al. on the existence of heterochromatic cycles of length 3 or 4.
Let $K_n$ be the complete graph with $n$ vertices and $c_1, c_2, ..., c_r$ be $r$ different colors. Suppose we randomly and uniformly color the edges of $K_n$ in $c_1, c_2, ..., c_r$. Then we get a random graph, denoted by $\mathcal{K}_n^r$. In the paper, we investigate the asymptotic properties of several kinds of monochromatic and heterochromatic subgraphs in $\mathcal{K}_n^r$. Accurate threshold functions in some cases are also obtained.
In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e., heterochromatic triangle free complete graphs; the other is heterochromatic triangle free graphs with $k$-good colorings, i.e., minimum color degree at least $k$. For the heterochromatic triangle free graphs $K_n$, we obtain that for every vertex $v\in V(K_n)$, $K_n$ has a heterochromatic $v$-path of length at least $d^c(v)$; whereas for the heterochromatic …
In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose $f$-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is $f$-chromatic if each color $c$ appears on at most $f(c)$ edges. We also present a necessary and sufficient condition for edge-colored graphs to have an $f$-chromatic spanning forest with exactly $m$ components. Moreover, usin…
In 2001, Brualdi and Hollingsworth proved that an edge-colored balanced complete bipartite graph Kn,n with a color set C = {1,2,3,..., 2n-1} has a heterochromatic spanning tree if the number of edges colored with colors in R is more than |R|^2 /4 for any non-empty subset R \subseteq C, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors, namely, any color appears at most once. In 2010, Suzuki generalized heterochromatic graphs to f-chromatic graphs, where any color c appears at most f(c). Moreover, he presented a necessary and sufficient condition for grap…
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 $G$ 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…
A heterochromatic (or rainbow) graph is an edge-colored graph whose edges have distinct colors, that is, where each color appears at most once. In this paper, I propose a $(g,f)$-chromatic graph as an edge-colored graph where each color $c$ appears at least $g(c)$ times and at most $f(c)$ times. I also present a necessary and sufficient condition for edge-colored graphs (not necessary to be proper) to have a $(g,f)$-chromatic spanning tree. Using this criterion, I show that an edge-colored complete graph $G$ has a spanning tree with a color probability distribution `similar' to that of $G$. Mo…
Let $G$ be an edge colored graph. A {\it}{rainbow path} in $G$ is a path in which all the edges are colored with distinct colors. Let $d^c(v)$ be the color degree of a vertex $v$ in $G$, i.e. the number of distinct colors present on the edges incident on the vertex $v$. Let $t$ be the maximum length of a rainbow path in $G$. Chen and Li showed that if $d^c \geq k$, for every vertex $v$ of $G$, then $t \geq \left \lceil \frac{3 k}{5}\right \rceil + 1$ (Long heterochromatic paths in edge-colored graphs, The Electronic Journal of Combinatorics 12 (2005), # R33, Pages:1-33.) Unfortunately, proof b…
We prove a quantitative version of the multi-colored Motzkin-Rabin theorem in the spirit of [BDWY12]: Let $V_1,\ldots,V_n \subset R^d$ be $n$ disjoint sets of points (of $n$ `colors'). Suppose that for every $V_i$ and every point $v \in V_i$ there are at least $δ|V_i|$ other points $u \in V_i$ so that the line connecting $v$ and $u$ contains a third point of another color. Then the union of the points in all $n$ sets is contained in a subspace of dimension bounded by a function of $n$ and $δ$ alone.
Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain the exact value of the Gallai-Ramsey numbers for the union of two stars in many cases and bounds in other cases. This work represents the first class of disconnected graphs to be considered as the desired monochromatic subgraph.
Let $S$ and $T$ be subsets of $\mathbf{F}_q^n$. We show there are subsets $S'$ of $S$ and $T'$ of $T$ such that $S+T$ is the union of $S+T'$ and $S'+T$, with $|S'| + |T'|$ bounded by $c^n$ with $c < q$. The proof relies on the method of Croot-Lev-Pach and Ellenberg-Gijswijt on the cap set problem, together with a result of Meshulam on linear spaces of low-rank matrices. The result is a modest generalization of the recent bounds on (single-colored and multi-colored) sum-free sets by the author and others.
We introduce the concept of complete edge-colored permutation graphs as complete graphs that are the edge-disjoint union of "classical" permutation graphs. We show that a graph $G=(V,E)$ is a complete edge-colored permutation graph if and only if each monochromatic subgraph of $G$ is a "classical" permutation graph and $G$ does not contain a triangle with~$3$ different colors. Using the modular decomposition as a framework we demonstrate that complete edge-colored permutation graphs are characterized in terms of their strong prime modules, which induce also complete edge-colored permutation gr…
An edge-colored graph is called rainbow if all the colors on its edges are distinct. Given a positive integer n and a graph G, the anti-Ramsey number ar(n,G) is the maximum number of colors in an edge-coloring of K_{n} with no rainbow copy of G. Denote by kC_{3} the union of k vertex-disjoint copies of C_{3}. In this paper, we determine the anti-Ramsey number ar(n,kC_{3}) for n=3k and n\geq2k^{2}-k+2, respectively. When 3k\leq n\leq 2k^{2}-k+2, we give lower and upper bounds for ar(n, kC_{3}).
Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for double stars $S(n,m)$, where $S(n,m)$ is the graph obtained from the union of two stars $K_{1,n}$ and $K_{1,m}$ by adding an edge between their centers. We also provide the sharp result in some cases.
In FOCS 2002, Even et al. showed that any set of $n$ discs in the plane can be Conflict-Free colored with a total of at most $O(\log n)$ colors. That is, it can be colored with $O(\log n)$ colors such that for any (covered) point $p$ there is some disc whose color is distinct from all other colors of discs containing $p$. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: \begin{enumerate} \item [(i)] Any set of $n$ discs in the plane can be colored with a total of at most $O(k \log n)$ colors such that (a) for any point $p$ that is…
Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex 4-colorings of triangulations of the 2-sphere to finding such colorings for the class \(\mathfrak H\) of triangulations of the 2-sphere that have a Hamiltonian circuit. This has been used by Whitney and others from 1936 to the present to find equivalent reformulations of the 4 Color Theorem (4CT). Recently there has been activity to try to use some of these reformuations to find a shorter proof of the 4CT. Every triangulation in \(\mathfrak H\) has a dual graph that is a union of two binary trees with the same number …
Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer k, find a set of at least k pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains NP-hard but admits constant-ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-unio…
A path-matching of order $p$ is a vertex disjoint union of nontrivial paths spanning $p$ vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers $r, p_1, \dots, p_r$, define $R^{PM}(p_1, \dots, p_r)$ to be the smallest integer $n$ such that in any $r$-coloring of the edges of $K_n$ there exists a path-matching of color $i$ and order at least $p_i$ for some $i\in [r]$. Our main result is that for $r\geq 2$ and $p_1\geq \dots\geq p_r\geq 2$, if $p_1\ge…
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息力学学院）；LIRMM, Univ Montpellier, CNRS（蒙彼利埃大学CNRS LIRMM实验室）；KAIST, School of Computing（韩国科学技术院计算机学院）
