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 The Dominator Edge Coloring of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Minhui Li ◽  
Shumin Zhang ◽  
Caiyun Wang ◽  
Chengfu Ye
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Simple Graph ◽  
Edge Colorings ◽  
Color Class ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Relationship Of ◽  
The Relationship

Let G be a simple graph. A dominator edge coloring (DE-coloring) of G is a proper edge coloring in which each edge of G is adjacent to every edge of some color class (possibly its own class). The dominator edge chromatic number (DEC-number) of G is the minimum number of color classes among all dominator edge colorings of G , denoted by χ d ′ G . In this paper, we establish the bounds of the DEC-number of a graph, present the DEC-number of special graphs, and study the relationship of the DEC-number between G and the operations of G .

The algorithm for adjacent vertex distinguishing proper edge coloring of graphs

2015 ◽  
Vol 07 (04) ◽  
pp. 1550044
Author(s):  
Jingwen Li ◽  
Tengyun Hu ◽  
Fei Wen
Keyword(s):  
Chromatic Number ◽  
Time Complexity ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Optimal Solutions ◽  
Test Results ◽  
Intelligent Algorithm ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Fixed Order

An adjacent vertex distinguishing proper edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that no pair of adjacent vertices meet the same set of colors. The minimum number of colors is called adjacent vertex distinguishing proper edge chromatic number of [Formula: see text]. In this paper, we present a new heuristic intelligent algorithm to calculate the adjacent vertex distinguishing proper edge chromatic number of graphs. To be exact, the algorithm establishes two objective subfunctions and a main objective function to find its optimal solutions by the conditions of adjacent vertex distinguishing proper edge coloring. Moreover, we test and analyze its feasibility, and the test results show that this algorithm can rapidly and efficiently calculate the adjacent vertex distinguishing proper edge chromatic number of graphs with fixed order, and its time complexity is less than [Formula: see text].

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

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.

scholarly journals Introduction to dominated edge chromatic number of a graph

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.

scholarly journals On The Local Edge Antimagic Coloring of Corona Product of Path and Cycle

CAUCHY ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 40
Author(s):  
Siti Aisyah ◽  
Ridho Alfarisi ◽  
Rafiantika M. Prihandini ◽  
Arika Indah Kristiana ◽  
Ratna Dwi Christyanti
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Edge Coloring ◽  
Antimagic Labeling ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Vertex Set ◽  
Local Edge ◽  
Corona Product

<p>Let  be a nontrivial and connected graph of vertex set  and edge set  . A bijection  is called a local edge antimagic labeling if for any two adjacent edges  and , where for . Thus, the local edge antimagic labeling induces a proper edge coloring of G if each edge e assigned the color  . The color of each an edge <em>e</em> = <em>uv</em> is assigned bywhich is defined by the sum of label both and vertices  and  . The local edge antimagic chromatic number, denoted by  is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of   . In our paper, we present the local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, cycle corona cycle.</p><p><strong>Keywords:</strong> Local antimagic; edge coloring; corona product; path; cycle.</p>

scholarly journals On the edge coloring of graph products

2005 ◽  
Vol 2005 (16) ◽  
pp. 2669-2676 ◽  
Author(s):  
M. M. M. Jaradat
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Sufficient Condition ◽  
Graph Products ◽  
Strong Product ◽  
Class 1 ◽  
Minimum Number

The edge chromatic number ofGis the minimum number of colors required to color the edges ofGin such a way that no two adjacent edges have the same color. We will determine a sufficient condition for a various graph products to be of class 1, namely, strong product, semistrong product, and special product.

ON THE CHROMATIC NUMBERS OF FUNCTIGRAPHS

2012 ◽  
Vol 13 (03n04) ◽  
pp. 1250011 ◽  
Author(s):  
GEORGE QI ◽  
SHENGHAO WANG ◽  
WEIZHEN GU
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Sufficient Conditions ◽  
Chromatic Numbers ◽  
Disjoint Copy ◽  
Minimum Number ◽  
Definition Of

The chromatic number of a graph G, denoted χ(G) is the minimum number of colors needed to color vertices of G so that no two adjacent vertices share the same color. A functigraph over a given graph is obtained as follows: Let G' be a disjoint copy of a given G and f be a function f : V(G) → V(G'). The functigraph over G, denoted by C(G, f), is the graph with V(C(G, f)) = V(G) ∪ V(G') and E(C(G, f)) = E(G) ∪ E(G') ∪ {uv : u ∈ V(G), v ∈ V(G'), v = f(u)}. Recently, Chen et al. proved that [Formula: see text]. In this paper, we first provide sufficient conditions on functions f to reach the lower bound for any graph. We then study the attainability of the chromatic numbers of functigraphs. Finally, we extend the definition of a functigraph in different ways and then investigate the bounds of chromatic numbers of such graphs.

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 $L(e)$.

Strong chromatic index of generalized Jahangir graphs and generalized Helm graphs

2021 ◽  
Author(s):  
Vikram Srinivasan Thiru ◽  
S. Balaji
Keyword(s):  
Edge Coloring ◽  
Chromatic Index ◽  
Proper Coloring ◽  
Strong Chromatic Index ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Different Color

The strong edge coloring of a graph G is a proper edge coloring that assigns a different color to any two edges which are at most two edges apart. The minimum number of color classes that contribute to such a proper coloring is said to be the strong chromatic index of G. This paper defines the strong chromatic index for the generalized Jahangir graphs and the generalized Helm graphs.

scholarly journals Acyclic, Star and Oriented Colourings of Graph Subdivisions

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Undirected Graph ◽  
The Other ◽  
Chromatic Index ◽  
Chromatic Numbers ◽  
Minimum Number ◽  
International Audience ◽  
Vertex Partitions ◽  
Colour Classes ◽  
Vertex Colouring

International audience Let G be a graph with chromatic number χ (G). A vertex colouring of G is \emphacyclic if each bichromatic subgraph is a forest. A \emphstar colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χ _a(G) and χ _s(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G' be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G' correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χ _a(G'), χ _s(G') and χ (G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The \emphoriented chromatic number χ ^→(G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that χ ^→(G')=χ (G) whenever χ (G)≥ 9.

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