On the Chromatic Number of Non-Sparse Random Intersection Graphs

2016 ◽  
Vol 60 (1) ◽  
pp. 112-127
Author(s):  
Sotiris E. Nikoletseas ◽  
Christoforos L. Raptopoulos ◽  
Paul G. Spirakis
Keyword(s):  
Chromatic Number ◽  
Intersection Graphs

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 $6k-6$.

scholarly journals Triangle-free intersection graphs of line segments with large chromatic number

2014 ◽  
Vol 105 ◽  
pp. 6-10 ◽  
Author(s):  
Arkadiusz Pawlik ◽  
Jakub Kozik ◽  
Tomasz Krawczyk ◽  
Michał Lasoń ◽  
Piotr Micek ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Intersection Graphs ◽  
Line Segments ◽  
Large Chromatic Number

scholarly journals Box and Segment Intersection Graphs with Large Girth and Chromatic Number

2021 ◽  
Author(s):  
James Davies
Keyword(s):  
Chromatic Number ◽  
Intersection Graphs ◽  
Large Girth ◽  
Straight Lines

We prove that there are intersection graphs of axis-aligned boxes in R3 and intersection graphs of straight lines in R3 that have arbitrarily large girth and chromatic number.

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.

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 Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

2013 ◽  
Vol 50 (3) ◽  
pp. 714-726 ◽  
Author(s):  
Arkadiusz Pawlik ◽  
Jakub Kozik ◽  
Tomasz Krawczyk ◽  
Michał Lasoń ◽  
Piotr Micek ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Intersection Graphs ◽  
Geometric Intersection Graphs ◽  
Large Chromatic Number

scholarly journals Burling Graphs, Chromatic Number, and Orthogonal Tree-Decompositions

10.37236/7052 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Stefan Felsner ◽  
Gwenaël Joret ◽  
Piotr Micek ◽  
William T. Trotter ◽  
Veit Wiechert
Keyword(s):  
Chromatic Number ◽  
Negative Answer ◽  
Intersection Graphs ◽  
Path Decomposition ◽  
Classic Result ◽  
Tree Decompositions ◽  
Path Decompositions ◽  
Large Chromatic Number

A classic result of Asplund and Grünbaum states that intersection graphs of axis-aligned rectangles in the plane are $\chi$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two path-decompositions such that each bag of the first decomposition intersects each bag of the second in at most $k$ vertices has chromatic number at most $f(k)$. Recently, Dujmović, Joret, Morin, Norin, and Wood asked whether this remains true more generally for two tree-decompositions. In this note we provide a negative answer: There are graphs with arbitrarily large chromatic number for which one can find two tree-decompositions such that each bag of the first decomposition intersects each bag of the second in at most two vertices. Furthermore, this remains true even if one of the two decompositions is restricted to be a path-decomposition. This is shown using a construction of triangle-free graphs with unbounded chromatic number due to Burling, which we believe should be more widely known.

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