On the vertex-distinguishing proper edge coloring of composition of complete graph and star

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

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NEIGHBOR SUM DISTINGUISHING COLORING OF SOME GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830912500474 ◽

2012 ◽

Vol 04 (04) ◽

pp. 1250047 ◽

Cited By ~ 16

Author(s):

AIJUN DONG ◽

GUANGHUI WANG

Keyword(s):

Planar Graph ◽

Edge Coloring ◽

Proper Edge Coloring

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ max {2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ max {2Δ(G), 19} if G is a graph such that mad(G) ≤ 5.

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Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500354 ◽

2020 ◽

Vol 12 (04) ◽

pp. 2050035

Author(s):

Danjun Huang ◽

Xiaoxiu Zhang ◽

Weifan Wang ◽

Stephen Finbow

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Edge Coloring ◽

Discrete Math ◽

Adjacent Vertex ◽

Chromatic Index ◽

Proper Edge Coloring ◽

Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

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Colorful Partitions of Cardinal Numbers

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-056-4 ◽

1979 ◽

Vol 31 (3) ◽

pp. 524-541

Author(s):

J. Baumgartner ◽

P. Erdös ◽

F. Galvin ◽

J. Larson

Keyword(s):

Complete Graph ◽

Edge Coloring ◽

Cardinal Numbers ◽

Symmetrizable Space ◽

Element Set

Use the two element subsets of κ, denoted by [κ]2, as the edge set for the complete graph on κ points. Write CP(κ, µ, v) if there is an edge coloring R: [κ]2 → µ with µ colors so that for every proper v element set X ⊊ κ, there is a point x ∈ κ ∼ X so that the edges between x and X receive at least the minimum of µ and v colors. Write CP⧣(K, µ, v) if the coloring is oneto- one on the edges between x and elements of X.Peter W. Harley III [5] introduced CP and proved that for κ ≧ ω, CP(κ+, κ, κ) holds to solve a topological problem, since the fact that CP(ℵ1, ℵ0, ℵ0) holds implies the existence of a symmetrizable space on ℵ1 points in which no point is a Gδ.

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

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.333-335.1452 ◽

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.

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Algorithms of Determining any Perfect Matching Mi of Kv

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.48-49.170 ◽

2011 ◽

Vol 48-49 ◽

pp. 170-173

Author(s):

Zhao Di Xu ◽

Xiao Yi Li ◽

Wan Xi Chou

Keyword(s):

Complete Graph ◽

Perfect Matching ◽

Edge Coloring ◽

Round Robin ◽

Matrix Algorithm

A definition about edge-matrix is given. Two algorithms for solving perfect matching are obtained. Algorithms A is that perfect matching is determined by using edge coloring of edge-matrix ; Algorithm B is that perfect matching is determined by partitioning edge-matrix into sub matrix and also by solving perfect matching of a complete graph .The procedure of constructing round-robin tournament by using algorithm A and round-robin tournament by using algorithm B.

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Introduction to dominated edge chromatic number of a graph

Opuscula Mathematica ◽

10.7494/opmath.2021.41.2.245 ◽

2021 ◽

Vol 41 (2) ◽

pp. 245-257

Author(s):

Mohammad R. Piri ◽

Saeid Alikhani

Keyword(s):

Chromatic Number ◽

Edge Coloring ◽

Color Class ◽

Proper Edge Coloring ◽

Minimum Number

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$, is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by $\chi_{dom}^{\prime}(G)$. We obtain some properties of $\chi_{dom}^{\prime}(G)$ and compute it for specific graphs. Also examine the effects on $\chi_{dom}^{\prime}(G)$, when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivision of $G$ and study the dominated edge chromatic number of these kind of graphs.

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The Rainbow Connection Number of Triangular Snake Graphs

10.5753/etc.2020.11091 ◽

2020 ◽

Author(s):

Aleffer Rocha ◽

Sheila M. Almeida ◽

Leandro M. Zatesko

Keyword(s):

Information Security ◽

Edge Coloring ◽

Connection Number ◽

Rainbow Connection Number ◽

Proper Edge Coloring ◽

Rainbow Connection ◽

Rainbow Coloring

Rainbow coloring problems, of noteworthy applications in Information Security, have been receiving much attention last years in Combinatorics. The rainbow connection number of a graph G is the least number of colors for a (not necessarily proper) edge coloring of G such that between any pair of vertices there is a path whose edge colors are all distinct. In this paper we determine the rainbow connection number of the triple triangular snake graphs.

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A Note on Path Factors of $(3,4)$-Biregular Bipartite Graphs

The Electronic Journal of Combinatorics ◽

10.37236/705 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 2

Author(s):

Carl Johan Casselgren

Keyword(s):

Graph Theory ◽

Bipartite Graph ◽

Edge Coloring ◽

Bipartite Graphs ◽

The Other ◽

Spanning Subgraph ◽

Interval Coloring ◽

Proper Edge Coloring

A proper edge coloring of a graph $G$ with colors $1,2,3,\dots$ is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is $(3,4)$-biregular if all vertices in one part have degree $3$ and all vertices in the other part have degree $4$. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph $G$ has a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in $\{2, 4, 6, 8\}$, then $G$ has an interval coloring. It was also conjectured that every simple $(3,4)$-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple $(3,4)$-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at $3$-valent vertices and lengths not exceeding $22$.

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Upper density of monochromatic infinite paths

Advances in Combinatorics ◽

10.19086/aic.10810 ◽

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.

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