Rainbow Matching in Edge-Colored Graphs

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The color degree of a vertex $v$ is the number of different colors on edges incident to $v$. Wang and Li conjectured that for $k\geq 4$, every edge-colored graph with minimum color degree at least $k$ contains a rainbow matching of size at least $\left\lceil k/2 \right\rceil$. We prove the slightly weaker statement that a rainbow matching of size at least $\left\lfloor k/2 \right\rfloor$ is guaranteed. We also give sufficient conditions for a rainbow matching of size at least $\left\lceil k/2 \right\rceil$ that fail to hold only for finitely many exceptions (for each odd $k$).

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Rainbow Path and Color Degree in Edge Colored Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3770 ◽

2014 ◽

Vol 21 (1) ◽

Cited By ~ 2

Author(s):

Anita Das ◽

S. V. Subrahmanya ◽

P. Suresh

Keyword(s):

Short Proof ◽

Maximum Length ◽

Colored Graph ◽

New Approach ◽

Colored Graphs ◽

Degree Of A Vertex ◽

Color Degree ◽

Rainbow Path

Let $G$ be an edge colored graph. A rainbow pathin $G$ 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 (2005) showed that if $d^c \geq k \,\, (k\geq 8)$, for every vertex $v$ of $G$, then $t \geq \left \lceil \frac{3 k}{5}\right \rceil + 1$. Unfortunately, the proof by Chen and Li is very long and comes to about 23 pages in the journal version. Chen and Li states in their paper that it was conjectured by Akira Saito, that $t \ge \left \lceil \frac {2k} {3} \right \rceil$. They also state in their paper that they believe $t \ge k - c$ for some constant $c$. In this note, we give a short proof to show that $t \ge \left \lceil \frac{3 k}{5}\right \rceil$, using an entirely different method. Our proof is only about 2 pages long. The draw-back is that our bound is less by 1, than the bound given by Chen and Li. We hope that the new approach adopted in this paper would eventually lead to the settlement of the conjectures by Saito and/or Chen and Li.

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Long Heterochromatic Paths in Edge-Colored Graphs

The Electronic Journal of Combinatorics ◽

10.37236/1930 ◽

2005 ◽

Vol 12 (1) ◽

Cited By ~ 9

Author(s):

He Chen ◽

Xueliang Li

Keyword(s):

Lower Bound ◽

Colored Graph ◽

Degree Condition ◽

Colored Graphs ◽

Degree Of A Vertex ◽

Color Degree

Let $G$ be an edge-colored graph. A heterochromatic path of $G$ is such a path in which no two edges have the same color. $d^c(v)$ denotes the color degree of a vertex $v$ of $G$. In a previous paper, we showed that if $d^c(v)\geq k$ for every vertex $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lceil{k+1\over 2}\rceil$. It is easy to see that if $k=1,2$, $G$ has a heterochromatic path of length at least $k$. Saito conjectured that under the color degree condition $G$ has a heterochromatic path of length at least $\lceil{2k+1\over 3}\rceil$. Even if this is true, no one knows if it is a best possible lower bound. Although we cannot prove Saito's conjecture, we can show in this paper that if $3\leq k\leq 7$, $G$ has a heterochromatic path of length at least $k-1,$ and if $k\geq 8$, $G$ has a heterochromatic path of length at least $\lceil{3k\over 5}\rceil+1$. Actually, we can show that for $1\leq k\leq 5$ any graph $G$ under the color degree condition has a heterochromatic path of length at least $k$, with only one exceptional graph $K_4$ for $k=3$, one exceptional graph for $k=4$ and three exceptional graphs for $k=5$, for which $G$ has a heterochromatic path of length at least $k-1$. Our experience suggests us to conjecture that under the color degree condition $G$ has a heterochromatic path of length at least $k-1$.

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Some algorithmic results for finding compatible spanning circuits in edge-colored graphs

Journal of Combinatorial Optimization ◽

10.1007/s10878-020-00644-7 ◽

2020 ◽

Vol 40 (4) ◽

pp. 1008-1019

Author(s):

Zhiwei Guo ◽

Hajo Broersma ◽

Ruonan Li ◽

Shenggui Zhang

Keyword(s):

Polynomial Time ◽

Sufficient Conditions ◽

Complete Graphs ◽

Hamilton Cycles ◽

Colored Graph ◽

Colored Graphs ◽

Complete Problem ◽

Polynomial Time Algorithms ◽

Np Complete ◽

Euler Tours

Abstract A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.

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Rainbow Matchings of size $m$ in Graphs with Total Color Degree at least $2mn$

The Electronic Journal of Combinatorics ◽

10.37236/8239 ◽

2020 ◽

Vol 27 (3) ◽

Author(s):

Jürgen Kritschgau

Keyword(s):

Proper Coloring ◽

Colored Graphs ◽

Rainbow Matching ◽

Color Degree ◽

Host Graph

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.

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Rainbow Matchings of Size $\delta(G)$ in Properly Edge-Colored Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2443 ◽

2012 ◽

Vol 19 (2) ◽

Cited By ~ 10

Author(s):

Jennifer Diemunsch ◽

Michael Ferrara ◽

Allan Lo ◽

Casey Moffatt ◽

Florian Pfender ◽

...

Keyword(s):

Minimum Degree ◽

Time Algorithm ◽

Colored Graph ◽

Colored Graphs ◽

Rainbow Matching ◽

Delta G

A 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 $f(\delta)$ such that a properly edge-colored graph $G$ with minimum degree $\delta$ and order at least $f(\delta)$ must have a rainbow matching of size $\delta$. We answer this question in the affirmative; an extremal approach yields that $f(\delta) = 98\delta/23< 4.27\delta$ suffices. Furthermore, we give an $O(\delta(G)|V(G)|^2)$-time algorithm that generates such a matching in a properly edge-colored graph of order at least $6.5\delta$.

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Rainbow Matchings in Properly Edge Colored Graphs

The Electronic Journal of Combinatorics ◽

10.37236/649 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 13

Author(s):

Guanghui Wang

Keyword(s):

Minimum Degree ◽

Colored Graph ◽

Free Graph ◽

Colored Graphs ◽

Rainbow Matching

Let $G$ be a properly edge colored graph. A rainbow matching of $G$ is a matching in which no two edges have the same color. Let $\delta$ denote the minimum degree of $G$. We show that if $|V(G)|\geq \frac{8\delta}{5}$, then $G$ has a rainbow matching of size at least $\lfloor\frac {3 \delta }{5}\rfloor$. We also prove that if $G$ is a properly colored triangle-free graph, then $G$ has a rainbow matching of size at least $\lfloor\frac {2 \delta }{3}\rfloor$.

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Heterochromatic Matchings in Edge-Colored Graphs

The Electronic Journal of Combinatorics ◽

10.37236/862 ◽

2008 ◽

Vol 15 (1) ◽

Cited By ~ 12

Author(s):

Guanghui Wang ◽

Hao Li

Keyword(s):

Bipartite Graph ◽

Colored Graph ◽

Colored Graphs ◽

Color Degree

Let $G$ be an (edge-)colored graph. A heterochromatic matching of $G$ is a matching in which no two edges have the same color. For a vertex $v$, let $d^c(v)$ be the color degree of $v$. We show that if $d^c(v)\geq k$ for every vertex $v$ of $G$, then $G$ has a heterochromatic matching of size $\big\lceil{5k-3\over 12}\big\rceil$. For a colored bipartite graph with bipartition $(X,Y)$, we prove that if it satisfies a Hall-like condition, then it has a heterochromatic matching of cardinality $\big\lceil{|X|\over 2}\big\rceil$, and we show that this bound is best possible.

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The Action of the Weyl Group on the $$E_8$$ Root System

Graphs and Combinatorics ◽

10.1007/s00373-021-02315-8 ◽

2021 ◽

Author(s):

Rosa Winter ◽

Ronald van Luijk

Keyword(s):

Automorphism Group ◽

Weyl Group ◽

Root System ◽

Sufficient Conditions ◽

Maximal Clique ◽

Necessary And Sufficient Conditions ◽

Inner Product ◽

Colored Graphs ◽

Necessary And Sufficient ◽

Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

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On paths, trails and closed trails in edge-colored graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.586 ◽

2012 ◽

Vol Vol. 14 no. 2 (Graph Theory) ◽

Author(s):

Laurent Gourvès ◽

Adria Lyra ◽

Carlos A. Martinhon ◽

Jérôme Monnot

Keyword(s):

Polynomial Time ◽

Optimal Solution ◽

Colored Graph ◽

Edge Disjoint ◽

Colored Graphs ◽

International Audience ◽

Np Complete ◽

T Path ◽

Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

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