On the clique number of the square of a line graph and its relation to maximum degree of the line graph

2019 ◽  
Vol 92 (3) ◽  
pp. 261-274 ◽  
Author(s):  
Maxime Faron ◽  
Luke Postle
Keyword(s):  
Maximum Degree ◽  
Line Graph ◽  
Clique Number

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

scholarly journals Line Graphs of Monogenic Semigroup Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Nihat Akgunes ◽  
Yasar Nacaroglu ◽  
Sedat Pak
Keyword(s):  
Maximum Degree ◽  
Zero Divisor ◽  
Minimum Degree ◽  
Line Graph ◽  
Domination Number ◽  
Clique Number ◽  
Line Graphs ◽  
Monogenic Semigroup ◽  
Graph Properties ◽  
Graph Parameters

The concept of monogenic semigroup graphs Γ S M is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L Γ S M of Γ S M . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L Γ S M .

scholarly journals An upper bound for the chromatic number of line graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Andrew D. King ◽  
Bruce A. Reed ◽  
Adrian R. Vetta
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Upper Bound ◽  
Maximum Degree ◽  
Line Graph ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Line Graphs ◽  
International Audience

International audience It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.

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.

scholarly journals The resolving number of a graph Delia

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Delia Garijo ◽  
Antonio González ◽  
Alberto Márquez
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Clique Number ◽  
Metric Dimension ◽  
Np Hard ◽  
Arbitrary Graph ◽  
International Audience ◽  
Graph Parameters ◽  
Two Parameters

Graph Theory International audience We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.

scholarly journals Squared Chromatic Number Without Claws or Large Cliques

2019 ◽  
Vol 62 (1) ◽  
pp. 23-35
Author(s):  
Wouter Cames van Batenburg ◽  
Ross J. Kang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Clique Number ◽  
Sharp Bound ◽  
Line Graphs ◽  
Free Graph ◽  
Simple Graphs ◽  
Strengthened Form

AbstractLet $G$ be a claw-free graph on $n$ vertices with clique number $\unicode[STIX]{x1D714}$, and consider the chromatic number $\unicode[STIX]{x1D712}(G^{2})$ of the square $G^{2}$ of $G$. Writing $\unicode[STIX]{x1D712}_{s}^{\prime }(d)$ for the supremum of $\unicode[STIX]{x1D712}(L^{2})$ over the line graphs $L$ of simple graphs of maximum degree at most $d$, we prove that $\unicode[STIX]{x1D712}(G^{2})\leqslant \unicode[STIX]{x1D712}_{s}^{\prime }(\unicode[STIX]{x1D714})$ for $\unicode[STIX]{x1D714}\in \{3,4\}$. For $\unicode[STIX]{x1D714}=3$, this implies the sharp bound $\unicode[STIX]{x1D712}(G^{2})\leqslant 10$. For $\unicode[STIX]{x1D714}=4$, this implies $\unicode[STIX]{x1D712}(G^{2})\leqslant 22$, which is within 2 of the conjectured best bound. This work is motivated by a strengthened form of a conjecture of Erdős and Nešetřil.

scholarly journals On Bounding the Difference of the Maximum Degree and the Clique Number

2014 ◽  
Vol 31 (5) ◽  
pp. 1689-1702 ◽  
Author(s):  
Oliver Schaudt ◽  
Vera Weil
Keyword(s):  
Maximum Degree ◽  
Clique Number ◽  
The Difference

scholarly journals Expanders Are Universal for the Class of All Spanning Trees

2013 ◽  
Vol 22 (2) ◽  
pp. 253-281 ◽  
Author(s):  
DANIEL JOHANNSEN ◽  
MICHAEL KRIVELEVICH ◽  
WOJCIECH SAMOTIJ
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Spanning Trees ◽  
Clique Number ◽  
Interesting Consequence ◽  
Random Regular Graph ◽  
Vertex Graph ◽  
Graph Class ◽  
Universal Graphs ◽  
Natural Expansion

A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex graph satisfying certain natural expansion properties is(n,Δ)-universal. Our methods also apply to the case when Δ is some function ofn. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constantcsuch that the random graphG(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degreecn2/3log2n. Another interesting consequence is the existence of locally sparsen-vertex(n,Δ)-universal graphs. For example, we show that one can (randomly) constructn-vertex(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose(n,Δ)-universal graphs with merelyO(n)edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly(n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

scholarly journals A Note on Coloring Vertex-Transitive Graphs

10.37236/4626 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Clique Number ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\}$, and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $\Delta \ge 13$ we prove the Borodin–Kostochka conjecture, i.e., $\chi\le\max\{\omega,\Delta-1\}$.

scholarly journals Colouring Squares of Claw-free Graphs

2019 ◽  
Vol 71 (1) ◽  
pp. 113-129 ◽  
Author(s):  
Rémi de Joannis de Verclos ◽  
Ross J. Kang ◽  
Lucas Pastor
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Clique Number ◽  
Simple Graph ◽  
General Question ◽  
Free Graph ◽  
Original Question

AbstractIs there some absolute $\unicode[STIX]{x1D700}>0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\unicode[STIX]{x1D712}(G^{2})\leqslant (2-\unicode[STIX]{x1D700})\unicode[STIX]{x1D714}(G)^{2}$, where $\unicode[STIX]{x1D714}(G)$ is the clique number of $G$? Erdős and Nešetřil asked this question for the specific case where $G$ is the line graph of a simple graph, and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and, moreover, that it essentially reduces to the original question of Erdős and Nešetřil.

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