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.

Generalized edge-colorings of weighted graphs

2016 ◽  
Vol 08 (01) ◽  
pp. 1650015
Author(s):  
Yuji Obata ◽  
Takao Nishizeki
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Edge Coloring ◽  
Efficient Algorithms ◽  
Upper And Lower Bounds ◽  
Weighted Graphs ◽  
Chromatic Index ◽  
Edge Colorings ◽  
Integer Weight

Let [Formula: see text] be a graph with a positive integer weight [Formula: see text] for each vertex [Formula: see text]. One wishes to assign each edge [Formula: see text] of [Formula: see text] a positive integer [Formula: see text] as a color so that [Formula: see text] for any vertex [Formula: see text] and any two edges [Formula: see text] and [Formula: see text] incident to [Formula: see text]. Such an assignment [Formula: see text] is called an [Formula: see text]-edge-coloring of [Formula: see text], and the maximum integer assigned to edges is called the span of [Formula: see text]. The [Formula: see text]-chromatic index of [Formula: see text] is the minimum span over all [Formula: see text]-edge-colorings of [Formula: see text]. In the paper, we present various upper and lower bounds on the [Formula: see text]-chromatic index, and obtain three efficient algorithms to find an [Formula: see text]-edge-coloring of a given graph. One of them finds an [Formula: see text]-edge-coloring with span smaller than twice the [Formula: see text]-chromatic index.

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

2013 ◽  
Vol 333-335 ◽  
pp. 1452-1455
Author(s):  
Chun Yan Ma ◽  
Xiang En Chen ◽  
Fang Yang ◽  
Bing Yao
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

scholarly journals Upper density of monochromatic infinite paths

2019 ◽  
Author(s):  
Jan Corsten ◽  
Louis DeBiasio ◽  
Ander Lamaison ◽  
Richard Lang
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Classical Case ◽  
Infinite Graphs ◽  
Edge Colorings ◽  
Vertex Set ◽  
Upper Density ◽  
Countably Infinite ◽  
Positive Integers ◽  
Red Path

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

scholarly journals On the simultaneous edge coloring of graphs

2014 ◽  
Vol 06 (04) ◽  
pp. 1450049
Author(s):  
Behrooz Bagheri Gh. ◽  
Behnaz Omoomi
Keyword(s):  
Bipartite Graph ◽  
Edge Coloring ◽  
Close Relation ◽  
Bipartite Graphs ◽  
Double Cover ◽  
Edge Colorings ◽  
Graphic Sequence

A μ-simultaneous edge coloring of graph G is a set of μ proper edge colorings of G with a same color set such that for each vertex, the sets of colors appearing on the edges incident to that vertex are the same in each coloring and no edge receives the same color in any two colorings. The μ-simultaneous edge coloring of bipartite graphs has a close relation with μ-way Latin trades. Mahdian et al. (2000) conjectured that every bridgeless bipartite graph is 2-simultaneous edge colorable. Luo et al. (2004) showed that every bipartite graphic sequence S with all its elements greater than one, has a realization that admits a 2-simultaneous edge coloring. In this paper, the μ-simultaneous edge coloring of graphs is studied. Moreover, the properties of the extremal counterexample to the above conjecture are investigated. Also, a relation between 2-simultaneous edge coloring of a graph and a cycle double cover with certain properties is shown and using this relation, some results about 2-simultaneous edge colorable graphs are obtained.

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 Remarks on upper and lower bounds formatching sequencibility of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2091-2099
Author(s):  
Shuya Chiba ◽  
Yuji Nakano
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Hamilton Cycle ◽  
Upper And Lower Bounds ◽  
Linear Ordering ◽  
The Relationship

In 2008, Alspach [The Wonderful Walecki Construction, Bull. Inst. Combin. Appl. 52 (2008) 7-20] defined the matching sequencibility of a graph G to be the largest integer k such that there exists a linear ordering of its edges so that every k consecutive edges in the linear ordering form a matching of G, which is denoted by ms(G). In this paper, we show that every graph G of size q and maximum degree ? satisfies 1/2?q/?+1? ? ms(G) ? ?q?1/??1? by using the edge-coloring of G, and we also improve this lower bound for some particular graphs. We further discuss the relationship between the matching sequencibility and a conjecture of Seymour about the existence of the kth power of a Hamilton cycle.

scholarly journals List Edge-Colorings of Series-Parallel Graphs

10.37236/1474 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Martin Juvan ◽  
Bojan Mohar ◽  
Robin Thomas
Keyword(s):  
Maximum Degree ◽  
Edge Coloring ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Parallel Graph

It is proved that for every integer $k\ge3$, for every (simple) series-parallel graph $G$ with maximum degree at most $k$, and for every collection $(L(e):e\in E(G))$ of sets, each of size at least $k$, there exists a proper edge-coloring of $G$ such that for every edge $e\in E(G)$, the color of $e$ belongs to $L(e)$.

scholarly journals On Edge Colorings with at Least q Colors in Every Subset of p Vertices

10.37236/1553 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Gábor N. Sárközy ◽  
Stanley Selkow
Keyword(s):  
Edge Coloring ◽  
Log P ◽  
Q Value ◽  
P Values ◽  
Edge Colorings

For fixed integers $p$ and $q$, an edge coloring of $K_n$ is called a $(p, q)$-coloring if the edges of $K_n$ in every subset of $p$ vertices are colored with at least $q$ distinct colors. Let $f(n, p, q)$ be the smallest number of colors needed for a $(p, q)$-coloring of $K_n$. In [3] Erdős and Gyárfás studied this function if $p$ and $q$ are fixed and $n$ tends to infinity. They determined for every $p$ the smallest $q$ ($= {p \choose 2} - p + 3$) for which $f(n,p,q)$ is linear in $n$ and the smallest $q$ for which $f(n,p,q)$ is quadratic in $n$. They raised the question whether perhaps this is the only $q$ value which results in a linear $f(n,p,q)$. In this paper we study the behavior of $f(n,p,q)$ between the linear and quadratic orders of magnitude. In particular we show that that we can have at most $\log p$ values of $q$ which give a linear $f(n,p,q)$.

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