scholarly journals The fractional chromatic number, the Hall ratio, and the lexicographic product

2009 ◽  
Vol 309 (14) ◽  
pp. 4746-4749 ◽  
Author(s):  
P.D. Johnson
Keyword(s):  
Chromatic Number ◽  
Lexicographic Product ◽  
Fractional Chromatic Number

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 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.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

On the b-chromatic number of some graph products

2012 ◽  
Vol 49 (2) ◽  
pp. 156-169 ◽  
Author(s):  
Marko Jakovac ◽  
Iztok Peterin
Keyword(s):  
Chromatic Number ◽  
Upper And Lower Bounds ◽  
Vertex Coloring ◽  
Graph Products ◽  
Lexicographic Product ◽  
Arbitrary Graph ◽  
Strong Product ◽  
Color Class ◽  
Complete Bipartite Graphs ◽  
Proper Vertex

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G ⊠ H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of Pn ⊠ H, Cn ⊠ H, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

scholarly journals The fractional chromatic number of Zykov products of graphs

2011 ◽  
Vol 24 (4) ◽  
pp. 432-437 ◽  
Author(s):  
Pierre Charbit ◽  
Jean Sébastien Sereni
Keyword(s):  
Chromatic Number ◽  
Fractional Chromatic Number

scholarly journals The Non-Crossing Graph

10.37236/1140 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Nathan Linial ◽  
Michael Saks ◽  
David Statter
Keyword(s):  
Chromatic Number ◽  
Independence Number ◽  
Clique Number ◽  
The Other ◽  
Fractional Chromatic Number ◽  
Vertex Set

Two sets are non-crossing if they are disjoint or one contains the other. The non-crossing graph ${\rm NC}_n$ is the graph whose vertex set is the set of nonempty subsets of $[n]=\{1,\ldots,n\}$ with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover number of this graph are presented. For the chromatic number of this graph we show: $$ n(\log_e n -\Theta(1)) \le \chi({\rm NC}_n) \le n (\lceil\log_2 n\rceil-1). $$

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