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

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

Discrete Mathematics ◽

10.1016/j.disc.2020.112042 ◽

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

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Decomposing edge-colored graphs under color degree constraints

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2017.06.078 ◽

2017 ◽

Vol 61 ◽

pp. 491-497 ◽

Cited By ~ 2

Author(s):

Shinya Fujita ◽

Ruonan Li ◽

Guanghui Wang

Keyword(s):

Colored Graphs ◽

Color Degree

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

The Electronic Journal of Combinatorics ◽

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 $t_3(K(n,n))=3$.

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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 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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Rainbow Matching in Edge-Colored Graphs

The Electronic Journal of Combinatorics ◽

10.37236/475 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 15

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 $k$).

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