scholarly journals On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane

10.37236/1805 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Alexandr Kostochka ◽  
Kittikorn Nakprasit
Keyword(s):  
Chromatic Number ◽  
Convex Sets ◽  
Convex Set ◽  
Clique Number ◽  
Finite Family ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Homothetic Copies

Let $G$ be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane. We show that every such graph with clique number $k$ is $(3k-3)$-degenerate. This bound is sharp. As a consequence, we derive that $G$ is $(3k-2)$-colorable. We show also that the chromatic number of every intersection graph $H$ of a family of homothetic copies of a fixed convex set in the plane with clique number $k$ is at most $6k-6$.

SOME RESULTS ON THE INTERSECTION GRAPHS OF IDEALS OF RINGS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250200 ◽  
Author(s):  
S. AKBARI ◽  
R. NIKANDISH ◽  
M. J. NIKMEHR
Keyword(s):  
Chromatic Number ◽  
Clique Number ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Graph Theoretic ◽  
Reduced Ring ◽  
Algebraic Properties ◽  
Vertex Set

Let R be a ring with unity and I(R)* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)* and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals is finite. Among other results, it is shown that for a ring R, if the clique number of G(R) is finite, then the chromatic number is finite and if R is a reduced ring, then both are equal.

ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A PARTIALLY ORDERED SET

2017 ◽  
Vol 97 (2) ◽  
pp. 185-193 ◽  
Author(s):  
SARIKA DEVHARE ◽  
VINAYAK JOSHI ◽  
JOHN LAGRANGE
Keyword(s):  
Partially Ordered Set ◽  
Zero Divisor ◽  
Ordered Set ◽  
Clique Number ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Partially Ordered ◽  
Zero Divisor Graph

In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero divisor graph of a lattice’,Bull. Aust. Math. Soc. 89(2014), 177–190]. As a consequence, the intersection graph of an intersection-closed family of nonempty subsets of a set is weakly perfect if it has finite clique number. These results are applied to annihilating-ideal graphs and intersection graphs of submodules.

scholarly journals Intersection graphs of modules and rings

2018 ◽  
Vol 17 (07) ◽  
pp. 1850131
Author(s):  
Jerzy Matczuk ◽  
Marta Nowakowska ◽  
Edmund R. Puczyłowski
Keyword(s):  
Structure Theorem ◽  
Clique Number ◽  
Intersection Graph ◽  
Chromatic Numbers ◽  
Finite Degree ◽  
Intersection Graphs ◽  
Complement Graph

We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in [S. Akbari, R. Nikandish and J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl. 12 (2013) 1250200]. We also generalize this result to modules, i.e. we get the structure theorem of modules for which their intersection graphs are infinite and contain maximal submodules of finite degree. Furthermore, we omit the assumption of maximality of submodules and still get a satisfactory characterization of such modules. In addition, we show that if the intersection graph of a module is infinite but its clique number is finite, then the clique and chromatic numbers of the graph coincide. This fact was known earlier only in some particular cases. It appears that such equality holds also in the complement graph.

INTERSECTION GRAPH OF SUBMODULES OF A MODULE

2012 ◽  
Vol 11 (01) ◽  
pp. 1250019 ◽  
Author(s):  
S. AKBARI ◽  
H. A. TAVALLAEE ◽  
S. KHALASHI GHEZELAHMAD
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Clique Number ◽  
Simple Graph ◽  
Intersection Graph ◽  
Star Graph ◽  
Graph Theoretic ◽  
Nonzero Intersection ◽  
One To One

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-moduleM, denoted by G(M), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonzero intersection. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star graph.

scholarly journals Restricted Frame Graphs and a Conjecture of Scott

10.37236/4424 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jérémie Chalopin ◽  
Louis Esperet ◽  
Zhentao Li ◽  
Patrice Ossona de Mendez
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraph ◽  
Intersection Graphs

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once.It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once.  We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.

scholarly journals Large Cliques in Sparse Random Intersection Graphs

10.37236/6233 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Mindaugas Bloznelis ◽  
Valentas Kurauskas
Keyword(s):  
Degree Distribution ◽  
Maximum Clique ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Intersection Graph ◽  
Optimal Order ◽  
Intersection Graphs ◽  
Random Intersection Graph ◽  
Finite Set

An intersection graph defines an adjacency relation between subsets $S_1,\dots, S_n$ of a finite set $W=\{w_1,\dots, w_m\}$: the subsets $S_i$ and $S_j$ are adjacent if they intersect. Assuming that the subsets are drawn independently at random according to the probability distribution $\mathbb{P}(S_i=A)=P(|A|){\binom{m}{|A|}}^{-1}$, $A\subseteq W$, where $P$ is a probability on $\{0, 1, \dots, m\}$, we obtain the random intersection graph $G=G(n,m,P)$. We establish  the asymptotic order of the clique number $\omega(G)$ of  a sparse random intersection graph as $n,m\to+\infty$. For $m = \Theta(n)$ we show that the maximum clique is of size $(1-\alpha/2)^{-\alpha/2}n^{1-\alpha/2}(\ln n)^{-\alpha/2}(1+o_P(1))$ in the case where the asymptotic degree distribution of $G$ is a power-law with exponent $\alpha \in (1,2)$. It is of size $\frac {\ln n} {\ln \ln n}(1+o_P(1))$ if the degree distribution has bounded variance, i.e., $\alpha>2$. We construct a simple polynomial-time algorithm which finds a clique of the optimal order $\omega(G) (1-o_P(1))$.

scholarly journals Turán-type results for intersection graphs of boxes

2021 ◽  
pp. 1-6
Author(s):  
István Tomon ◽  
Dmitriy Zakharov
Keyword(s):  
Short Note ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Poset Dimension ◽  
Axis Parallel ◽  
Turán Theorem ◽  
Separation Dimension

Abstract In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of K t,t , then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

A Krasnosel’skii-type theorem for certain orthogonal polytopes starshaped via k-staircase paths

2020 ◽  
Vol 20 (2) ◽  
pp. 169-177 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Type Theorem ◽  
Common Point ◽  
Finite Family ◽  
Intersection Graph ◽  
Axis Parallel ◽  
Orthogonal Polytopes

AbstractLet 𝒞 be a finite family of distinct axis-parallel boxes in ℝd whose intersection graph is a tree, and let S = ⋃{C : C in 𝒞}. If every two points of S see a common point of S via k-staircase paths, then S is starshaped via k-staircase paths. Moreover, the k-staircase kernel of S will be convex via k-staircases.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
Author(s):  
T. ASIR ◽  
T. TAMIZH CHELVAM
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Intersection Graph ◽  
Vertex Connectivity ◽  
Total Graph ◽  
Graph Theoretic ◽  
Vertex Set ◽  
Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

scholarly journals On the Isolated Vertices and Connectivity in Random Intersection Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graphs ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Random Intersection Graph ◽  
Probability Of Connectivity ◽  
Asymptotic Probability

We study isolated vertices and connectivity in the random intersection graph . A Poisson convergence for the number of isolated vertices is determined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When and , we give the asymptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in Erdős-Rényi random graphs.

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