Graphs with large clique-chromatic numbers

2015 ◽  
Vol 07 (04) ◽  
pp. 1550055
Author(s):  
Tanawat Wichianpaisarn ◽  
Chariya Uiyyasathian
Keyword(s):  
Chromatic Number ◽  
Maximal Clique ◽  
Line Graphs ◽  
Chromatic Numbers ◽  
Bounded Set ◽  
Initial Graph ◽  
The Family ◽  
Large Clique

The clique-chromatic number of a graph [Formula: see text], [Formula: see text], is the least number of colors on [Formula: see text] without a monocolored maximal clique of size at least two. If [Formula: see text] is triangle-free, [Formula: see text]; we then consider only graphs with a triangle. Unlike the chromatic number, the clique-chromatic number of a graph is not necessary to be at least those of its subgraphs. Thus, for any family of graphs [Formula: see text], the boundedness of [Formula: see text][Formula: see text] has been investigated. Many families of graphs are proved to have a bounded set of clique-chromatic numbers. In literature, only few families of graphs are shown to have an unbounded set of clique-chromatic numbers, for instance, the family of line graphs. This paper gives another family of graphs with such an unbounded set. These graphs are obtained by the well-known Mycielski’s construction with a certain property of the initial graph.

r-Dynamic Chromatic Number of Some Line Graphs

2018 ◽  
Vol 49 (4) ◽  
pp. 591-600 ◽  
Author(s):  
Hanna Furmańczyk ◽  
J. Vernold Vivin ◽  
N. Mohanapriya
Keyword(s):  
Chromatic Number ◽  
Line Graphs

scholarly journals Clique Colourings of Geometric Graphs

10.37236/7159 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Colin McDiarmid ◽  
Dieter Mitsche ◽  
Pawel Prałat
Keyword(s):  
Asymptotic Behaviour ◽  
Chromatic Number ◽  
Euclidean Distance ◽  
High Probability ◽  
Maximal Clique ◽  
Higher Dimensions ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Vertex Set

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number.  Given $n$ points $\mathbf{x}_1, \ldots,\mathbf{x}_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $\mathbf{x}_i$ and $\mathbf{x}_j$ is at most $r$. We investigate the clique chromatic number of such graphs.We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph $\mathcal{G}$ in the plane, where $n$ random points are independently and uniformly distributed in a suitable square. We see that as $r$ increases from 0, with high probability the clique chromatic number is 1 for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and finally drops back to 2.

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.

On the Babai and Upper Chromatic Numbers of Graphs of Diameter 2

2021 ◽  
Vol 52 (1) ◽  
pp. 113-123
Author(s):  
Peter Johnson ◽  
Alexis Krumpelman
Keyword(s):  
Chromatic Number ◽  
Metric Space ◽  
Simple Graph ◽  
Chromatic Numbers

The Babai numbers and the upper chromatic number are parameters that can be assigned to any metric space. They can, therefore, be assigned to any connected simple graph. In this paper we make progress in the theory of the first Babai number and the upper chromatic number in the simple graph setting, with emphasis on graphs of diameter 2.

scholarly journals On the Chromatic Thresholds of Hypergraphs

2015 ◽  
Vol 25 (2) ◽  
pp. 172-212
Author(s):  
JÓZSEF BALOGH ◽  
JANE BUTTERFIELD ◽  
PING HU ◽  
JOHN LENZ ◽  
DHRUV MUBAYI
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Open Problems ◽  
Fibre Bundles ◽  
Uniform Hypergraphs ◽  
The Family ◽  
Turán Number ◽  
Chromatic Thresholds ◽  
Extremal Set

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

2013 ◽  
Vol 333-335 ◽  
pp. 1452-1455
Author(s):  
Chun Yan Ma ◽  
Xiang En Chen ◽  
Fang Yang ◽  
Bing Yao
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

PARALLEL VERTEX COLOURING OF INTERVAL GRAPHS

1999 ◽  
Vol 10 (01) ◽  
pp. 19-31 ◽  
Author(s):  
G. SAJITH ◽  
SANJEEV SAXENA
Keyword(s):  
Chromatic Number ◽  
Interval Graph ◽  
Interval Graphs ◽  
Chromatic Numbers ◽  
Erew Pram ◽  
Pram Model ◽  
Interval Representation ◽  
Left And Right ◽  
Crcw Pram ◽  
Vertex Colouring

Evidence is given to suggest that minimally vertex colouring an interval graph may not be in NC 1. This is done by showing that 3-colouring a linked list is NC 1-reducible to minimally colouring an interval graph. However, it is shown that an interval graph with a known interval representation and an O(1) chromatic number can be minimally coloured in NC 1. For the CRCW PRAM model, an o( log n) time, polynomial processors algorithm is obtained for minimally colouring an interval graph with o( log n) chromatic number and a known interval representation. In particular, when the chromatic number is O(( log n)1-ε), 0<ε<1, the algorithm runs in O( log n/ log log n) time. Also, an O( log n) time, O(n) cost, EREW PRAM algorithm is found for interval graphs of arbitrary chromatic numbers. The following lower bound result is also obtained: even when the left and right endpoints of the interval are separately sorted, minimally colouring an interval graph needs Ω( log n/ log log n) time, on a CRCW PRAM, with a polynomial number of processors.

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