scholarly journals On the fractional chromatic number and the lexicographic product of graphs

1998 ◽  
Vol 185 (1-3) ◽  
pp. 259-263 ◽  
Author(s):  
Sandi Klavẑar
Keyword(s):  
Chromatic Number ◽  
Lexicographic Product ◽  
Fractional Chromatic Number ◽  
Product Of Graphs

Star coloring under some graph operations

2021 ◽  
pp. 2250079
Author(s):  
B. Akhavan Mahdavi ◽  
M. Tavakoli ◽  
F. Rahbarnia ◽  
Alireza Ashrafi
Keyword(s):  
Chromatic Number ◽  
Lexicographic Product ◽  
Proper Coloring ◽  
Sharp Bounds ◽  
Graph Operations ◽  
Join Of Graphs ◽  
Star Coloring ◽  
Product Of Graphs

A star coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] such that no path of length 3 in [Formula: see text] is bicolored. In this paper, the star chromatic number of join of graphs is computed. Some sharp bounds for the star chromatic number of corona, lexicographic, deleted lexicographic and hierarchical product of graphs together with a conjecture on the star chromatic number of lexicographic product of graphs are also presented.

scholarly journals Cubical coloring — fractional covering by cuts and semidefinite programming

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Robert Šámal
Keyword(s):  
Semidefinite Programming ◽  
Chromatic Number ◽  
Fractional Chromatic Number ◽  
Maximum Cut ◽  
Continuous Mappings ◽  
Polynomial Time Approximation Algorithm ◽  
Eigenvalue Bound ◽  
International Audience ◽  
Graph Parameters ◽  
Fractional Covering

International audience We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

scholarly journals Another H-super magic decompositions of the lexicographic product of graphs

2018 ◽  
Vol 2 (2) ◽  
pp. 72
Author(s):  
H Hendy ◽  
Kiki A. Sugeng ◽  
A.N.M Salman ◽  
Nisa Ayunda
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graph Decomposition ◽  
Graph Labeling ◽  
Sufficient Condition ◽  
Lexicographic Product ◽  
Simple Graphs ◽  
Product Of Graphs

<p>Let <span class="math"><em>H</em></span> and <span class="math"><em>G</em></span> be two simple graphs. The concept of an <span class="math"><em>H</em></span>-magic decomposition of <span class="math"><em>G</em></span> arises from the combination between graph decomposition and graph labeling. A decomposition of a graph <span class="math"><em>G</em></span> into isomorphic copies of a graph <span class="math"><em>H</em></span> is <span class="math"><em>H</em></span>-magic if there is a bijection <span class="math"><em>f</em> : <em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>) → {1, 2, ..., ∣<em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>)∣}</span> such that the sum of labels of edges and vertices of each copy of <span class="math"><em>H</em></span> in the decomposition is constant. A lexicographic product of two graphs <span class="math"><em>G</em><sub>1</sub></span> and <span class="math"><em>G</em><sub>2</sub>, </span> denoted by <span class="math"><em>G</em><sub>1</sub>[<em>G</em><sub>2</sub>], </span> is a graph which arises from <span class="math"><em>G</em><sub>1</sub></span> by replacing each vertex of <span class="math"><em>G</em><sub>1</sub></span> by a copy of the <span class="math"><em>G</em><sub>2</sub></span> and each edge of <span class="math"><em>G</em><sub>1</sub></span> by all edges of the complete bipartite graph <span class="math"><em>K</em><sub><em>n</em>, <em>n</em></sub></span> where <span class="math"><em>n</em></span> is the order of <span class="math"><em>G</em><sub>2</sub>.</span> In this paper we provide a sufficient condition for <span class="math">$\overline{C_{n}}[\overline{K_{m}}]$</span> in order to have a <span class="math">$P_{t}[\overline{K_{m}}]$</span>-magic decompositions, where <span class="math"><em>n</em> &gt; 3, <em>m</em> &gt; 1, </span> and <span class="math"><em>t</em> = 3, 4, <em>n</em> − 2</span>.</p>

scholarly journals On the set chromatic number of the join and comb product of graphs

2020 ◽  
Vol 1538 ◽  
pp. 012009
Author(s):  
B C L Felipe ◽  
A D Garciano ◽  
M A C Tolentino
Keyword(s):  
Chromatic Number ◽  
Product Of Graphs

scholarly journals Total Coloring Conjecture for Certain Classes of Graphs

Algorithms ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 161 ◽  
Author(s):  
R. Vignesh ◽  
J. Geetha ◽  
K. Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Product Line ◽  
Total Coloring ◽  
Lexicographic Product ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

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