scholarly journals Long Heterochromatic Paths in Edge-Colored Graphs

10.37236/1930 ◽  
2005 ◽  
Vol 12 (1) ◽  
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 $k-1$.

scholarly journals Rainbow Path and Color Degree in Edge Colored Graphs

10.37236/3770 ◽  
2014 ◽  
Vol 21 (1) ◽  
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.

scholarly journals Rainbow Matching in Edge-Colored Graphs

10.37236/475 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Timothy D. LeSaulnier ◽  
Christopher Stocker ◽  
Paul S. Wenger ◽  
Douglas B. West
Keyword(s):  
Sufficient Conditions ◽  
Colored Graph ◽  
Colored Graphs ◽  
Rainbow Matching ◽  
Degree Of A Vertex ◽  
Color Degree

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 $k$).

scholarly journals Partitioning 3-Edge-Colored Complete Equi-Bipartite Graphs by Monochromatic Trees under a Color Degree Condition

10.37236/855 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Xueliang Li ◽  
Fengxia Liu
Keyword(s):  
Bipartite Graphs ◽  
Complete Multipartite Graph ◽  
Colored Graph ◽  
Degree Condition ◽  
Partition Number ◽  
Color Degree ◽  
Minimum Integer ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Vertex Disjoint

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 $t_2(K(n_1,n_2,\cdots,n_k))$. 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 $t_3(K(n,n))=3$.

scholarly journals Color Neighborhood Union Conditions for Long Heterochromatic Paths in Edge-Colored Graphs

10.37236/995 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
He Chen ◽  
Xueliang Li
Keyword(s):  
Lower Bound ◽  
Colored Graph ◽  
Colored Graphs ◽  
Union Condition ◽  
Neighborhood Union Condition ◽  
Color Neighborhood ◽  
Neighborhood Union

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 $CN(v)$ denote the color neighborhood of a vertex $v$ of $G$. In a previous paper, we showed that 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 $\lfloor{2s+4\over5}\rfloor$. In the present paper, we prove that $G$ has a heterochromatic path of length at least $\lceil{s+1\over2}\rceil$, and give examples to show that the lower bound is best possible in some sense.

scholarly journals Heterochromatic Matchings in Edge-Colored Graphs

10.37236/862 ◽  
2008 ◽  
Vol 15 (1) ◽  
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.

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

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.

Color degree sum conditions for properly colored spanning trees in edge-colored graphs

2020 ◽  
Vol 343 (11) ◽  
pp. 112042
Author(s):  
Mikio Kano ◽  
Shun-ichi Maezawa ◽  
Katsuhiro Ota ◽  
Masao Tsugaki ◽  
Takamasa Yashima
Keyword(s):  
Spanning Trees ◽  
Degree Sum ◽  
Colored Graphs ◽  
Color Degree

scholarly journals Decomposing edge-colored graphs under color degree constraints

2017 ◽  
Vol 61 ◽  
pp. 491-497 ◽  
Author(s):  
Shinya Fujita ◽  
Ruonan Li ◽  
Guanghui Wang
Keyword(s):  
Colored Graphs ◽  
Color Degree

scholarly journals Genericity of chaos for colored graphs

2021 ◽  
Vol 9 (1) ◽  
pp. 37-52
Author(s):  
Ramón Barral Lijó ◽  
Hiraku Nozawa
Keyword(s):  
Colored Graph ◽  
Isomorphism Classes ◽  
Colored Graphs ◽  
Universal Space

Abstract To each colored graph one can associate its closure in the universal space of isomorphism classes of pointed colored graphs, and this subspace can be regarded as a generalized subshift. Based on this correspondence, we introduce two definitions for chaotic (colored) graphs, one of them analogous to Devaney’s. We show the equivalence of our two novel definitions of chaos, proving their topological genericity in various subsets of the universal space.

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