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 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 Induced Subgraphs of Graphs with Large Chromatic Number IX: Rainbow Paths

10.37236/6768 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Recent Result ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Large Chromatic Number ◽  
Vertex Colouring

We prove that for all integers $\kappa, s\ge 0$ there exists $c$ with the following property. Let $G$ be a graph with clique number at most $\kappa$ and chromatic number more than $c$. Then for every vertex-colouring (not necessarily optimal) of $G$, some induced subgraph of $G$ is an $s$-vertex path, and all its vertices have different colours. This extends a recent result of Gyárfás and Sárközy (2016) who proved the same for graphs $G$ with $\kappa=2$ and girth at least five.

scholarly journals On Graphs Related to Comaximal Ideals of a Commutative Ring

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Tongsuo Wu ◽  
Meng Ye ◽  
Dancheng Lu ◽  
Houyi Yu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraph ◽  
The Core ◽  
Maximal Ideals ◽  
Graph Properties ◽  
Vertex Set

We study the co maximal graph Ω(R), the induced subgraph Γ(R) of Ω(R) whose vertex set is R∖(U(R)∪J(R)), and a retract Γr(R) of Γ(R), where R is a commutative ring. For a graph Γ(R) which contains a cycle, we show that the core of Γ(R) is a union of triangles and rectangles, while a vertex in Γ(R) is either an end vertex or a vertex in the core. For a nonlocal ring R, we prove that both the chromatic number and clique number of Γ(R) are identical with the number of maximal ideals of R. A graph Γr(R) is also introduced on the vertex set {Rx∣x∈R∖(U(R)∪J(R))}, and graph properties of Γr(R) are studied.

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 Exact Distance Colouring in Trees

2018 ◽  
Vol 28 (2) ◽  
pp. 177-186 ◽  
Author(s):  
NICOLAS BOUSQUET ◽  
LOUIS ESPERET ◽  
ARARAT HARUTYUNYAN ◽  
RÉMI DE JOANNIS DE VERCLOS
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Short Note ◽  
Clique Number ◽  
Negative Answer ◽  
Bounded Degree ◽  
Study Interval ◽  
Simple Corollary ◽  
Bounded Degree Trees

For an integer q ⩾ 2 and an even integer d, consider the graph obtained from a large complete q-ary tree by connecting with an edge any two vertices at distance exactly d in the tree. This graph has clique number q + 1, and the purpose of this short note is to prove that its chromatic number is Θ((d log q)/log d). It was not known that the chromatic number of this graph grows with d. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant C such that for any odd integer d, any planar graph can be coloured with at most C colours such that any pair of vertices at distance exactly d have distinct colours. Finally, we study interval colouring of trees (where vertices at distance at least d and at most cd, for some real c > 1, must be assigned distinct colours), giving a sharp upper bound in the case of bounded degree trees.

scholarly journals THE CHARACTER GRAPH OF A FINITE GROUP IS PERFECT

2020 ◽  
pp. 1-5
Author(s):  
MAHDI EBRAHIMI
Keyword(s):  
Finite Group ◽  
Chromatic Number ◽  
Clique Number ◽  
Perfect Graph ◽  
Induced Subgraph ◽  
Delta G ◽  
A Finite Group ◽  
Complex Characters

Abstract For a finite group G, let $\Delta (G)$ denote the character graph built on the set of degrees of the irreducible complex characters of G. A perfect graph is a graph $\Gamma $ in which the chromatic number of every induced subgraph $\Delta $ of $\Gamma $ equals the clique number of $\Delta $ . We show that the character graph $\Delta (G)$ of a finite group G is always a perfect graph. We also prove that the chromatic number of the complement of $\Delta (G)$ is at most three.

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

r-Hued coloring of planar graphs without short cycles

2020 ◽  
Vol 12 (03) ◽  
pp. 2050034
Author(s):  
Yuehua Bu ◽  
Xiaofang Wang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

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 On edge-intersection graphs of k-bend paths in grids

2010 ◽  
Vol Vol. 12 no. 1 ◽  
Author(s):  
Therese Biedl ◽  
Michal Stern
Keyword(s):  
Conflict Resolution ◽  
Planar Graph ◽  
Bipartite Graph ◽  
Outerplanar Graph ◽  
Line Graphs ◽  
Intersection Graphs ◽  
Grid Networks ◽  
Graph Classes ◽  
International Audience ◽  
Number Of Bends

International audience Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k. We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with -regular edge orientations.

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