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.

Uniquely Colourable Graphs with Large Girth

1976 ◽  
Vol 28 (6) ◽  
pp. 1340-1344 ◽  
Author(s):  
Béla Bollobás ◽  
Norbert Sauer
Keyword(s):  
Chromatic Number ◽  
Large Girth ◽  
Large Chromatic Number

Tutte [1], writing under a pseudonym, was the first to prove that a graph with a large chromatic number need not contain a triangle. The result was rediscovered by Zykov [5] and Mycielski [4]. Erdös [2] proved the much stronger result that for every k ≧ 2 and g there exist a k-chromatic graph whose girth is at least g.

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 List edge chromatic number of graphs with large girth

1992 ◽  
Vol 101 (1-3) ◽  
pp. 189-201 ◽  
Author(s):  
A.V. Kostochka
Keyword(s):  
Chromatic Number ◽  
Large Girth

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 On a conjecture concerning helly circle graphs

2003 ◽  
Vol 23 (1) ◽  
pp. 221-229 ◽  
Author(s):  
Guillermo Durán ◽  
Agustín Gravano ◽  
Marina Groshaus ◽  
Fábio Protti ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Systems Of Equations ◽  
Intersection Graphs ◽  
Circle Graph ◽  
Circle Graphs ◽  
Analytic Tool ◽  
Two Systems ◽  
Cartesian Plane ◽  
Definition Of ◽  
Straight Lines

We say that G is an e-circle graph if there is a bijection between its vertices and straight lines on the cartesian plane such that two vertices are adjacent in G if and only if the corresponding lines intersect inside the circle of radius one. This definition suggests a method for deciding whether a given graph G is an e-circle graph, by constructing a convenient system S of equations and inequations which represents the structure of G, in such a way that G is an e-circle graph if and only if S has a solution. In fact, e-circle graphs are exactly the circle graphs (intersection graphs of chords in a circle), and thus this method provides an analytic way for recognizing circle graphs. A graph G is a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point. A conjecture by Durán (2000) states that G is a Helly circle graph if and only if G is a circle graph and contains no induced diamonds (a diamond is a graph formed by four vertices and five edges). Many unsuccessful efforts - mainly based on combinatorial and geometrical approaches - have been done in order to validate this conjecture. In this work, we utilize the ideas behind the definition of e-circle graphs and restate this conjecture in terms of an equivalence between two systems of equations and inequations, providing a new, analytic tool to deal with it.

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 Circular Chromatic Number of Signed Graphs

10.37236/9938 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Reza Naserasr ◽  
Zhouningxin Wang ◽  
Xuding Zhu
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Simple Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Outerplanar Graphs ◽  
Circular Chromatic Number ◽  
Basic Properties ◽  
Large Girth ◽  
Sigma E

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph (loops and multi edges allowed) and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, \sigma)$ with no positive loop, a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$.  We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular,  we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Máčajová, Raspaud, and Škoviera. 

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.

Sign in / Sign up

Export Citation Format

Share Document