In the paper we state and prove theorem describing the upper bound on number of the graphs that have fixed number of vertices |V| and can be colored with the fixed number of n colors. The bound relates both numbers using power of 2, while the exponent is the difference between |V| and n. We also state three conjectures on the number of graphs that have fixed number of vertices |V| and chromatic number n.
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every edge-colored complete graph without containing monochromatic triangles always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the excepti…
A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a properly colored directed path in $D$. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.
Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-Häggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers $s$ and $t$ with $t\geq s\geq2$, every edge-colored graph $G$ with no properly colored $K_{s,t}$ contains a spanning subgraph $H$ which admits an orientation $D$ such that every directed cycle in $D$ is a properly colored cycle in $G$. Using this result, we show that for $r\geq4$, if the Caccetta-Hä…
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.
It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Δ^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker results for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain se…
In 2009, Blagojevic, Matschke & Ziegler established the first tight colored Tverberg theorem, but no lower bounds for the number of colored Tverberg partitions. We develop a colored version of our previous results (2008), and we extend our results from the uncolored version: Evenness and non-trivial lower bounds for the number of colored Tverberg partitions. This follows from similar results on the number of colored Birch partitions.
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.
It is shown that conditions stronger in a certain sense than color-shifting cannot be placed on the class of colored complexes without changing the characterization of the flag f-vectors.
In the paper Graphical complexity of products of permutation groups, M. Grech, A. Jez, A. Kisielewicz have proved that the direct product of automorphism groups of edge-colored graphs is itself the automorphism groups of an edge-colored graph. In this paper, we study the direct product of two permutation groups such that at least one of them fails to be the automorphism group of an edge-colored graph. We find necessary and sufficient conditions for the direct product to be the automorphism group of an edge-colored graph. The same problem is solved for the edge-colored digraphs.
The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$
Generalizing Reiner's notion of set partitions of type $B_n$, we define colored $B_n$-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored $B_n$-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored $B_n$-partition. We find an asymptotic expression of the total number of colored $B_n$-partitions up to an error of $O(n^{-1/2}\log^{7/2}{n})$, and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored $B_n$-pa…
In responding to a question on Math Stackexchange, the author formulated the problem of determining the number of strings of balls colored in most $n$ colors with a number $k$ of repeated colors. In this paper, we formulate the problem more formally, and solve several variations.
Given an integer $1\leq j <n$, define the $(j)$-coloring of a $n$-dimensional hypercube $H_{n}$ to be the $2$-coloring of the edges of $H_{n}$ in which all edges in dimension $i$, $1\leq i \leq j$, have color $1$ and all other edges have color $2$. Cheng et al. [Proper distance in edge-colored hypercubes, Applied Mathematics and Computation 313 (2017) 384-391.] determined the number of distinct shortest properly colored paths between a pair of vertices for the $(1)$-colored hypercubes. It is natural to consider the number for $(j)$-coloring, $j\geq 2$. In this note, we determine the number of …
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…
Recent investigations in computational biology focus on a family of 2-colored digraphs, called 2-colored best match graphs, which naturally arise from rooted phylogenetic trees. Actually the defining properties of such graphs are unexpectedly unusual in graph theory, and they were established only recently after the discovery of their links to evolutionary relatedness via phylogenetic trees. In this paper several results are obtained on 2-colored best match graphs which well fit in the mainstream of graph theory.
We exhibit a one-to-one correspondence between $3$-colored graphs and subarrangements of certain hyperplane arrangements denoted $\mathcal J_n$, $n \in \mathbb N$. We define the notion of centrality of $3$-colored graphs which corresponds to the centrality of hyperplane arrangements. Via the correspondence, the characteristic polynomial $χ_{\mathcal J_n}$ of $\mathcal J_n$ can be expressed in terms of the number of central $3$-colored graphs, and we compute $χ_{\mathcal J_n}$ for $n = 2, 3$.
We enumerate injectively $k$-colored rooted forests with a given number of vertices of each color and a given sequence of root colors. We obtain from this result some new multi-parameter distributions of Fuss-Catalan numbers. As an additional application we enumerate triangulations of regular convex polygons according to their proper 3-coloring type.
For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then $V(G)$ can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least $2$. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.
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 …
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…
We complete the characterization of the digraphs $D$ for which the induced $D$-removal lemma has polynomial bounds, answering a question of Alon and Shapira. We also study the analogous problem for $k$-colored complete graphs. In particular, we prove a removal lemma with polynomial bounds for Gallai colorings.
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…
Recent investigations in computational biology have focused on a family of 2-colored digraphs, called 2-colored best match graphs, which naturally arise from rooted phylogenetic trees. Actually the defining properties of such graphs are unusual, and a natural question is whether they also have properties which well fit in structural graph theory. In this paper, we prove that some underlying oriented bipartite graphs of a 2-colored best match graph are acyclic and we point out that the arising topological ordering can efficiently be used for constructing new families of 2-colored best match gra…
We consider the existence of patterned Hamilton cycles in randomly colored random graphs. Given a string $Π$ over a set of colors $\{1,2,\ldots,r\}$, we say that a Hamilton cycle is $Π$-colored if the pattern repeats at intervals of length $|Π|$ as we go around the cycle. We prove a hitting time for the existence of such a cycle. We also prove a hitting time result for a related notion of $Π$-connected.
We generalize the results of Ksavrelof and Zeng about the multidistribution of the excedance number of $S_n$ with some natural parameters to the colored permutation group and to the Coxeter group of type $D$. We define two different orders on these groups which induce two different excedance numbers. Surprisingly, in the case of the colored permutation group, we get the same generalized formulas for both orders.
We generalize Björner and Stanley's poset of compositions to $m$-colored compositions. Their work draws many analogies between their (1-colored) composition poset and Young's lattice of partitions, including links to (quasi-)symmetric functions and representation theory. Here we show that many of these analogies hold for any number of colors. While many of the proofs for Björner and Stanley's poset were simplified by showing isomorphism with the subword order, we remark that with 2 or more colors, our posets are not isomorphic to a subword order.
In~\cite{Hartman:2014}, Hartman, Hubi\v cka and Ma\v sulović studied the hierarchy of morphism extension classes for finite $L$-colored graphs, that is, undirected graphs without loops where sets of colors selected from $L$ are assigned to vertices and edges. They proved that when $L$ is a linear order, the classes $MH_L$ and $HH_L$ coincide, and the same is true for vertex-uniform finite $L$-colored graphs when $L$ is a diamond. In this paper, we explore the same question for countably infinite $L$-colored graphs. We prove that $MH_L=HH_L$ if and only if $L$ is a linear order.
Given a palette of six colors, a colored cube is a cube where each face is colored with exactly one color and each color appears on some face. Starting with an arbitrary collection of unit length colored cubes, one can try to arrange a subset of the collection into an $n \times n \times n$ cube where each face is a single color. This is the Colored Cubes Puzzle. In this paper, we determine minimum size sets of cubes required to complete an $n \times n \times n$ cube's frame, its corners and edges. We answer this problem for all $n$, and in particular show that for $n \geq4$ one has the best po…
An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is a morphism $φ:V(G)\rightarrow V(H)$ such that each edge (resp. arc) of $G$ is mapped to an edge (resp. arc) of $H$ of the same color (and orientation). An $(m,n)$-colored-mixed graph $T$ is said to be $P_g^{(m, n)}$-universal if every graph in $P_g^{(m, n)}$ (the planar $(m, n)$-colored-mi…
We study the number of factorizations of a positive integer, where the parts of the factorization are of l different colors (or kinds). Recursive or explicit formulas are derived for the case of unordered and ordered, distinct and non-distinct factorizations with at most and exactly l colors.
When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results forcing rainbow triangles to appear, we consider similar conditions which force the existence of a properly colored copy of $C_{4}$.
In this paper, we consider categories with colored morphisms and functors such that morphisms assigned to morphisms with a common color have a common color. In this paper, we construct a morphism-colored functor such that any morphism-colored functor from a given small morphism-colored groupoid to any discrete morphism-colored category factors through it. We also apply the main result to a schemoid constructed from a Hamming scheme.
Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $Δ^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A properly colored cycle (path) in $K_{n}^{c}$ is a cycle (path) in which adjacent edges have distinct colors. B. Bollobás and P. Erdös (1976) proposed the following conjecture: if $Δ^{\mathrm{mon}} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if $Δ^{\mathrm{mon}} (K_{n}^{c})< \lf…
In this paper, we present some results related to Barany-Larman colored problem and The Zivaljevic and Vrecica colored Tverberg problem. We give an alternative proof for the Barany-Larman Conjecture for primes -1 and the optimal colored Tverberg theorem. Also, we prove generalizations of the Colored Tverberg theorem of Zivaljevic and Vrecica.
We introduce a colored coalescent process which recovers random colored genealogical trees. Here a colored genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertice coalesce. The rule that governs the change of color involves a parameter $x$. When $x=1/2$, the colored coalescent process can be derived from a variant of the Wright-Fisher model for a haploid population in population genetics. Explicit computations of the expectation and the cumulative distribution function of the coalesce…
We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous results for $b$-colored partitions, where each part occurs in $b$ colors; for $b$-colored partitions with odd parts (or distinct parts); for partitions where the part $k$ comes in $k$ colors; and, overpartitions.
The $Γ$-colored $d$-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify $Γ$-colored $d$-complete posets which specifies the structure of the associated representations. We show that finite $Γ$-colored $d$-complete posets are precisely the dominant minuscule heaps of J.R. Stembridge. These heaps are reformulations and extensions of the colored $d$-complete posets of R.A. Proctor. We also show that connected infinite $Γ$-colored $d$-complete posets are precisely order filters of the connected full hea…
We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \wr S_n and compute the distribution of these statistics on the elements in these groups. We use some combinatorial approaches, recurrences, and generating functions manipulations to obtain our results.
In this paper, we prove a version of the Colored Tverberg Theorem with new constraints on the faces, in which we limit the number of faces with each one of the colors.
Clique-width is a complexity measure of directed as well as undirected graphs. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss an extension of the notion of rank-width to edge-colored graphs. A C-colored graph is a graph where the arcs are colored with colors from the set C. There is not a natural notion of rank-width for C-colored graphs. We define two notions of rank-width for them, both based on a coding of C-colored graphs by edge-colored graphs wher…
An $(m, n)$-colored mixed graph is a graph having arcs of $m$ different colors and edges of $n$ different colors. A graph homomorphism of an $(m, n$)-colored mixed graph $G$ to an $(m, n)$-colored mixed graph $H$ is a vertex mapping such that if $uv$ is an arc (edge) of color $c$ in $G$, then $f(u)f(v)$ is also an arc (edge) of color $c$. The ($m, n)$-colored mixed chromatic number of an $(m, n)$-colored mixed graph $G$, introduced by Nešetřil and Raspaud [J. Combin. Theory Ser. B 2000] is the order (number of vertices) of the smallest homomorphic image of $G$. Later Bensmail, Duffy and Sen [G…
Above-Guarantee Algorithm for Properly Colored Trees
Northwestern Polytechnical University（西北工业大学）；Xi’an-Budapest Joint Research Center for Combinatorics（西安-布达佩斯组合学联合研究中心）；MTA-ELTE Matroid Optimization Research Group（MTA-ELTE拟阵优化研究组）；HUN-REN–ELTE Egerváry Research Group, Department of Operations Research, ELTE Eötvös Loránd University（匈牙利研究与创新网络-罗兰大学埃格瓦里研究组，运筹学系，罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与创新网络阿尔弗雷德·雷尼数学研究所）
This paper shows how the study of colored compositions of integers reveals some unexpected and original connection with the Invert operator. The Invert operator becomes an important tool to solve the problem of directly counting the number of colored compositions for any coloration. The interesting consequences arising from this relationship also give an immediate and simple criterion to determine whether a sequence of integers counts the number of some colored compositions. Applications to Catalan and Fibonacci numbers naturally emerge, allowing to clearly answer to some open questions. Moreo…
A $c$-edge-colored multigraph has each edge colored with one of the $c$ available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parameters such as the number of edges and the rainbow degree.
We present a tiling of more than 99.985698% of the Euclidean plane with six colors, reducing the previous record for uncovered fraction of the plane by about 12.8%. We also present a tiling of more than 95.99% of the plane with five colors. It is thus shown that any unit-distance graph of order at most 6992 and 24 in the plane can be properly 6-colored and 5-colored, respectively.
We first consider the following problem. We are given a fixed perfect matching $M$ of $[n]$ and we add random edges one at a time until there is a Hamilton cycle containing $M$. We show that w.h.p. the hitting time for this event is the same as that for the first time there are no isolated vertices in the graph induced by the random edges. We then use this result for the following problem. We generate random edges and randomly color them black or white. A path/cycle is said to \emph{zebraic} if the colors alternate along the path. We show that w.h.p. the hitting time for a zebraic Hamilton cyc…
The main result of this paper is a "colored Tverberg theorem for rainbow-unavoidable complexes". This theorem may be considered as a merging of two theorems: "Tverberg theorem for collectively unavoidable complexes" and "balanced colored Tverberg theorem". The main tool for the proof is discrete Morse theory.
The contents of this 98-page paper have been subsumed into the 191-page paper "A colored sl(N)-homology for links in S^3" (arXiv:0907.0695v1 [math.GT]), in which we further develop the theory and use it to construct a colored link homology.
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…
Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of regular colored graphs of fixed positive degree and perform their exact and asymptotic enumeration. In particular we show that the generating function of the family of graphs of fixed degree is an algebraic series with a positive radius of convergence, independant of the degree. We describe the singular behavior of this series near its dominant singularity, and…
