Sufficient conditions for the existence of perfect heterochromatic matchings in colored graphs

Long heterochromatic paths in heterochromatic triangle free graphs

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 …

A generalization of heterochromatic graphs

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…

An f-chromatic spanning forest of edge-colored complete bipartite graphs

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…

Rainbow triangles in edge-colored graphs

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…

Long rainbow path in properly edge-colored complete graphs

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…

$(g,f)$-Chromatic spanning trees and forests

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…

Perfect k-colored matchings and (k+2)-gonal tilings

We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.

A note on randomly colored matchings in random bipartite graphs

We are given a bipartite graph that contains at least one perfect matching and where each edge is colored from a set $Q=\{c_1,c_2,\ldots,c_q}\$. Let $Q_i=\set{e\in E(G):c(e)=c_i}$, where $c(e)$ denotes the color of $e$. The perfect matching color profile $mcp(G)$ is defined to be the set of vectors $(m_1,m_2,\ldots,m_q)\in [n]^q$ such that there exists a perfect matching $M$ such that $|M\cap Q_i|=m_i$. We give bounds on the matching color profile for a randomly colored random bipartite graph.

Rainbow path and color degree in edge colored graphs

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…

三边着色二分图中的几乎完美彩色匹配

Almost-perfect colorful matchings in three-edge-colored bipartite graphs

该研究证明了三边着色二分图中存在满足各颜色边数指定条件的几乎完美彩色匹配，验证了Ryser-Brualdi-Stein猜想的三色多重扩展版本，且所得和的上界为最优。

Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance

Given an edge-colored complete graph $K_n$ on $n$ vertices, a perfect (respectively, near-perfect) matching $M$ in $K_n$ with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of $K_n$ by circular distance, and we denote the resulting complete graph by $K^{\bullet}_n$. We show that when $K^{\bullet}_n$ has an even number of vertices, it contains a rainbow perfect matching if and only if $n=8k$ or $n=8k+2$, where $k$ is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known…

着色三进制分拆、多项式序列与完全数

Colored base-3 partitions, sequences of polynomials, and perfect numbers

受含偶完全数子序列的三进制着色分拆计数启发，研究者定义四变量多项式序列并研究其性质，探讨两类切比雪夫多项式相关子序列，还分析了具组合意义与特殊零点分布的单变量多项式序列。

The threshold for the full perfect matching color profile in a random coloring of random graphs

Consider a graph $G$ with a coloring of its edge set $E(G)$ from a set $Q = \set{c_1,c_2, \ldots, c_q}$. Let $Q_i$ be the set of all edges colored with $c_i$. Recently, Frieze defined a notion of the perfect matching color profile denoted by $\mcp(G)$, which is the set of vectors $(m_1, m_2, \ldots, m_q) \in [n]^q$ such that there exists a perfect matching $M$ in $G$ with $|Q_i \cap M| = m_i$ for all $i$. Let $\a_1, \a_2, \ldots, \a_q$ be positive constants such that $\sum_{i=1}^q \a_i = 1$. Let $G$ be the random bipartite graph $G_{n,n,p}$. Suppose the edges of $G$ are independently colored w…

边着色图中的奇数正常着色圈

Odd Properly Colored Cycles in Edge-Colored Graphs

本文研究边着色图中奇数正常着色圈的存在性判定，并作为副产品给出检测给定边集中具有偶数或奇数条边的图中完美匹配的方法。

Perfectly Colorable Graphs

We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $ω(H)$ many colors where $ω(H)$ is the clique number of $H$. A graph is perfectly colorable if it admits a perfect coloring. We show that the class of perfectly colorable graphs is exactly the class of perfect paw-free graphs. It follows that perfectly colorable graphs can be recognized and colored in linear time.

强边着色图中的彩虹 spanning 结构

Rainbow spanning structures in strongly edge-colored graphs

本文研究了强边着色图中彩虹哈密尔顿回路的存在条件，证明了在足够大的n下，满足最小度数要求的图中必然存在彩虹哈密尔顿回路，并探讨了彩虹路径和完美匹配的最优条件。

Almost color-balanced perfect matchings in color-balanced complete graphs

For a graph $G$ and a not necessarily proper $k$-edge coloring $c:E(G)\to \{ 1,\ldots,k\}$, let $m_i(G)$ be the number of edges of $G$ of color $i$, and call $G$ {\it color-balanced} if $m_i(G)=m_j(G)$ for every two colors $i$ and $j$. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly $n$-edge colored complete bipartite graph $K_{n,n}$ has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whethe…

Rainbow Matchings and Hamilton Cycles in Random Graphs

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)…

二色图中的2-可达子集

2-reachable subsets in two-colored graphs

该研究针对English等人关于2边着色鸡尾酒会图顶点覆盖的猜想，证明了将直径2子集替换为2-可达子集的放宽形式成立，得出n顶点2色鸡尾酒会图必含大小至少为n/2的单色2-可达子集且该界最优的结论。

超图中完美匹配的差异度

Discrepancies of perfect matchings in hypergraphs

该研究确定了 r 边着色 k-一致超图中高差异度完美匹配的最小度阈值，并借助新的多色超图族回答了关于 3-一致超图的开放问题。

具有规定颜色数量的匹配

Matchings with Prescribed Color Counts

本文研究了完全二部图在特定红蓝边染色条件下的完美匹配问题，并证明了一个相关有趣结果。

两类全新无限族完全图的列表边着色猜想

The List Edge-Coloring Conjecture for New Infinite Families

本文证明了两类全新无限族完全图的列表边着色猜想，通过互补模方法控制1-因子分解的Pfaffian符号，建立行列式-Pfaffian桥梁并发展带符号Burnside-Fourier方法，同时揭示了该方法在高层论证中的赋值障碍。

平面上彩色点集的彩虹多边形

Rainbow polygons for colored point sets in the plane

该研究针对平面彩色点集的完美彩虹多边形最小顶点数问题，确定了k≤7时的rb-index(k)值，给出k≥5时的上下界，并提出O(n log n)时间的构造算法。

Elegantly colored paths and cycles in edge colored random graphs

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…

A short proof of Kotzig's theorem

A new shortest proof of Kotzig's Theorem about graphs with unique perfect matching is presented in this paper. It is well known that Kotzig's theorem is a consequence of Yeo's Theorem about edge-colored graph without alternating cycle. We present a proof of Yeo's Theorem based on the same ideas as our proof of Kotzig's theorem.

Weighted words at degree two, II: flat partitions, regular partitions, and application to level one perfect crystals

In a recent work, Keith and Xiong gave a refinement of Glaisher's theorem by using a Sylvester-style bijection. In this paper, we introduce two families of colored partitions, flat and regular partitions, and generalize the bijection of Keith and Xiong to these partitions. We then state two results, the first at degree one, where partitions have parts with primary colors, and the second result at degree two for secondary-colored partitions, using the result of the first paper of this series. These results allow us to easily retrieve the Frenkel-Kac character formulas of level one standard modu…

稠密图中无大独立集的单色三角形铺砌

Monochromatic triangle-tilings in dense graphs without large independent sets

本文研究了稠密图中无大独立集的2边着色图的弱单色三角形铺砌问题，给出了渐近最优的铺砌大小阈值。

Rainbow Matchings: existence and counting

A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of occurrences of a color. We also consider two natural models of random edge-colored K_{n,n} and show that, if the number of colors is at least n, then there is with high probability a random matching. This in particular shows that almost every square matrix of order n in which every entry appears at most n times has a Latin transversal.

基于Tutte定理的混合布尔约束求解量子启发的完美匹配问题

Solving Quantum-Inspired Perfect Matching Problems via Tutte's Theorem-Based Hybrid Boolean Constraints

针对量子计算中边着色图的受约束完美匹配问题，提出基于Tutte定理与优化技术的新型混合布尔约束编码，实验证明其扩展性显著优于竞争方法，凸显了设计特定问题编码的必要性。

顶点颜色约束下受量子启发的完美匹配

Quantum-Inspired Perfect Matching under Vertex-Color Constraints

该研究提出并分析了边双色图上带顶点颜色约束的完美匹配存在性问题EXISTS-PMVC，证明了决策图约束版本的NP难性，给出对称约束有界颜色版本的复杂度结论与更优原生算法，为约束匹配和量子实验设计提供新思路。