scholarly journals Strong Cliques in Claw-Free Graphs

2021 ◽  
Author(s):  
Michał Dębski ◽  
Małgorzata Śleszyńska-Nowak
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Maximum Size ◽  
Chromatic Index ◽  
Free Graph ◽  
Famous Conjecture ◽  
Strong Chromatic Index

AbstractFor a graph G, $$L(G)^2$$ L ( G ) 2 is the square of the line graph of G – that is, vertices of $$L(G)^2$$ L ( G ) 2 are edges of G and two edges $$e,f\in E(G)$$ e , f ∈ E ( G ) are adjacent in $$L(G)^2$$ L ( G ) 2 if at least one vertex of e is adjacent to a vertex of f and $$e\ne f$$ e ≠ f . The strong chromatic index of G, denoted by $$s'(G)$$ s ′ ( G ) , is the chromatic number of $$L(G)^2$$ L ( G ) 2 . A strong clique in G is a clique in $$L(G)^2$$ L ( G ) 2 . Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdős and Nešetřil concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree $$\varDelta $$ Δ is at most $$\varDelta ^2 + \frac{1}{2}\varDelta $$ Δ 2 + 1 2 Δ . This result improves the only known result $$1.125\varDelta ^2+\varDelta $$ 1.125 Δ 2 + Δ , which is a bound for the strong chromatic index of claw-free graphs.

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 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 Star multigraphs with three vertices of maximum degree

1986 ◽  
Vol 100 (2) ◽  
pp. 303-317 ◽  
Author(s):  
A. G. Chetwynd ◽  
A. J. W. Hilton
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Special Kind ◽  
Chromatic Index ◽  
Simple Graphs ◽  
Multiple Edges ◽  
Image Position ◽  
Edge Colouring

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with each non-simple edge. An edge-colouring of a multigraph G is a map ø: E(G)→, where is a set of colours and E(G) is the set of edges of G, such that no two edges receiving the same colour have a vertex in common. The chromatic index, or edge-chromatic numberχ′(G) of G is the least value of || for which an edge-colouring of G exists. Generalizing a well-known theorem of Vizing [14], we showed in [6] that, for a star-multigraph G,where Δ(G) denotes the maximum degree (that is, the maximum number of edges incident with a vertex) of G. Star-multigraphs for which χ′(G) = Δ(G) are said to be Class 1, and otherwise they are Class 2.

A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs

2002 ◽  
Vol 11 (1) ◽  
pp. 103-111 ◽  
Author(s):  
VAN H. VU
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Present Author ◽  
Upper Bounds ◽  
Constant Factor ◽  
Chromatic Index ◽  
Sparse Graphs ◽  
Critical Graphs ◽  
Multiplicative Constant ◽  
List Chromatic Number

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

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.

A BOUND FOR THE CHROMATIC NUMBER OF (, GEM)-FREE GRAPHS

2019 ◽  
Vol 100 (2) ◽  
pp. 182-188
Author(s):  
KATHIE CAMERON ◽  
SHENWEI HUANG ◽  
OWEN MERKEL
Keyword(s):  
Chromatic Number ◽  
Maximum Size ◽  
Free Graph ◽  
Induced Subgraph ◽  
Subgraph Isomorphic

As usual, $P_{n}$ ($n\geq 1$) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$. A graph is called ($P_{5}$, gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$, $\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$. We show that $\unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$ for every ($P_{5}$, gem)-free graph $G$.

scholarly journals The Distance-t Chromatic Index of Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 90-101 ◽  
Author(s):  
TOMÁŠ KAISER ◽  
ROSS J. KANG
Keyword(s):  
Maximum Degree ◽  
Upper Bounds ◽  
The Other ◽  
Graph Colouring ◽  
Chromatic Index ◽  
Multiplicative Factor ◽  
Strong Chromatic Index ◽  
Absolute Positive Constant ◽  
Large Maximum ◽  
Fixed Positive Integer

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

scholarly journals A Stronger Bound for the Strong Chromatic Index

2017 ◽  
Vol 27 (1) ◽  
pp. 21-43 ◽  
Author(s):  
HENNING BRUHN ◽  
FELIX JOOS
Keyword(s):  
Maximum Degree ◽  
Type Inequality ◽  
Chromatic Index ◽  
Strong Chromatic Index ◽  
Large Maximum

We prove χ′s(G) ≤ 1.93 Δ(G)2 for graphs of sufficiently large maximum degree where χ′s(G) is the strong chromatic index of G. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where we are allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.

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.

The Chromatic Index of a Claw-Free Graph Whose Core has Maximum Degree $$2$$ 2

2014 ◽  
Vol 31 (4) ◽  
pp. 805-811 ◽  
Author(s):  
S. Akbari ◽  
M. Ghanbari ◽  
K. Ozeki
Keyword(s):  
Maximum Degree ◽  
Chromatic Index ◽  
Free Graph
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