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

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. 

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 $k$).

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

Color Degree Condition for Long Rainbow Paths in Edge-Colored Graphs

2015 ◽  
Vol 39 (1) ◽  
pp. 409-425
Author(s):  
He Chen ◽  
Xueliang Li
Keyword(s):  
Degree Condition ◽  
Colored Graphs ◽  
Color Degree

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.

Large rainbow matchings in semi-strong edge-colorings of graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850021
Author(s):  
Zemin Jin ◽  
Kun Ye ◽  
He Chen ◽  
Yuefang Sun
Keyword(s):  
Lower Bounds ◽  
Edge Coloring ◽  
Natural Generalization ◽  
Edge Colorings ◽  
Colored Graphs ◽  
Rainbow Matching

The lower bounds for the size of maximum rainbow matching in properly edge-colored graphs have been studied deeply during the last decades. An edge-coloring of a graph [Formula: see text] is called a strong edge-coloring if each path of length at most three is rainbow. Clearly, the strong edge-coloring is a natural generalization of the proper one. Recently, Babu et al. considered the problem in the strongly edge-colored graphs. In this paper, we introduce a semi-strong edge-coloring of graphs and consider the existence of large rainbow matchings in it.

scholarly journals Color degree and heterochromatic cycles in edge-colored graphs

2012 ◽  
Vol 33 (8) ◽  
pp. 1958-1964 ◽  
Author(s):  
Hao Li ◽  
Guanghui Wang
Keyword(s):  
Colored Graphs ◽  
Color Degree

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

10.37236/2443 ◽  
2012 ◽  
Vol 19 (2) ◽  
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 $6.5\delta$. 

Color degree and monochromatic degree conditions for short properly colored cycles in edge-colored graphs

2017 ◽  
Vol 87 (3) ◽  
pp. 362-373 ◽  
Author(s):  
Shinya Fujita ◽  
Ruonan Li ◽  
Shenggui Zhang
Keyword(s):  
Colored Graphs ◽  
Color Degree ◽  
Degree Conditions

scholarly journals Color degree and alternating cycles in edge-colored graphs

2009 ◽  
Vol 309 (13) ◽  
pp. 4349-4354 ◽  
Author(s):  
Guanghui Wang ◽  
Hao Li
Keyword(s):  
Colored Graphs ◽  
Color Degree
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