scholarly journals Conflict-Free Colouring of Graphs

2013 ◽  
Vol 23 (3) ◽  
pp. 434-448 ◽  
Author(s):  
ROMAN GLEBOV ◽  
TIBOR SZABÓ ◽  
GÁBOR TARDOS
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Domination Number ◽  
Vertex Graph ◽  
Points Of View

We study the conflict-free chromatic number χCFof graphs from extremal and probabilistic points of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number ann-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős–Rényi random graphG(n,p) and give the asymptotics forp= ω(1/n). We also show that forp≥ 1/2 the conflict-free chromatic number differs from the domination number by at most 3.

scholarly journals On the Domination Number of a Random Graph

10.37236/1581 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Ben Wieland ◽  
Anant P. Godbole
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Moment Method ◽  
High Probability ◽  
Domination Number ◽  
Infinite Graph ◽  
Cantelli Lemma ◽  
Second Moment ◽  
Second Moment Method ◽  
The Second Moment Method

In this paper, we show that the domination number $D$ of a random graph enjoys as sharp a concentration as does its chromatic number $\chi$. We first prove this fact for the sequence of graphs $\{G(n,p_n\},\; n\to\infty$, where a two point concentration is obtained with high probability for $p_n=p$ (fixed) or for a sequence $p_n$ that approaches zero sufficiently slowly. We then consider the infinite graph $G({\bf Z}^+, p)$, where $p$ is fixed, and prove a three point concentration for the domination number with probability one. The main results are proved using the second moment method together with the Borel Cantelli lemma.

On the double total dominator chromatic number of graphs

2021 ◽  
pp. 2150154
Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Domination Number ◽  
Vertex Coloring ◽  
Total Domination Number ◽  
Close Relationship ◽  
Minimum Number ◽  
Number Formula ◽  
Dominator Coloring ◽  
Proper Vertex

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

scholarly journals Supersaturation of even linear cycles in linear hypergraphs

2020 ◽  
Vol 29 (5) ◽  
pp. 698-721
Author(s):  
Tao Jiang ◽  
Liana Yepremyan
Keyword(s):  
Random Graph ◽  
Classical Result ◽  
Independent Interest ◽  
Multiplicative Constant ◽  
Vertex Graph ◽  
Linear Hypergraphs

AbstractA classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

scholarly journals Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

2018 ◽  
Vol 6 (1) ◽  
pp. 343-356
Author(s):  
K. Arathi Bhat ◽  
G. Sudhakara
Keyword(s):  
Chromatic Number ◽  
Perfect Matching ◽  
Domination Number ◽  
Perfect Matchings ◽  
Factor Graph ◽  
Vertex Set ◽  
Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

Clean graph of a ring

2020 ◽  
pp. 2150156
Author(s):  
Mohammad HABIBI ◽  
Ece YETKİN ÇELİKEL ◽  
Ci̇hat ABDİOĞLU
Keyword(s):  
Chromatic Number ◽  
Crucial Role ◽  
Independence Number ◽  
Domination Number ◽  
Clique Number ◽  
Idempotent Element ◽  
Nonzero Idempotent

Let [Formula: see text] be a ring (not necessarily commutative) with identity. The clean graph [Formula: see text] of a ring [Formula: see text] is a graph with vertices in form [Formula: see text], where [Formula: see text] is an idempotent and [Formula: see text] is a unit of [Formula: see text]; and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we focus on [Formula: see text], the subgraph of [Formula: see text] induced by the set [Formula: see text] is a nonzero idempotent element of [Formula: see text]. It is observed that [Formula: see text] has a crucial role in [Formula: see text]. The clique number, the chromatic number, the independence number and the domination number of the clean graph for some classes of rings are determined. Moreover, the connectedness and the diameter of [Formula: see text] are studied.

Zero-divisor graphs of matrices over lattices

2016 ◽  
Vol 08 (04) ◽  
pp. 1650060 ◽  
Author(s):  
Anagha Khiste ◽  
Vinayak Joshi
Keyword(s):  
Chromatic Number ◽  
Zero Divisor ◽  
Domination Number ◽  
Basic Properties ◽  
Zero Divisor Graph ◽  
Element Chain ◽  
Text Element ◽  
Triangulated Graph

In this paper, we study basic properties such as connectivity, diameter and girth of the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over a lattice [Formula: see text] with 0. Further, we consider the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We determine the number of vertices, degree of each vertex, domination number and edge chromatic number of [Formula: see text]. Also, we show that Beck’s Conjecture is true for [Formula: see text]. Further, we prove that [Formula: see text] is hyper-triangulated graph.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

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