Clique Chromatic Number Research Articles

The jump of the clique chromatic number of random graphs

AbstractThe clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Prałat noted that around the clique chromatic number of the random graph changes by when we increase the edge‐probability by , but left the details of this surprising “jump” phenomenon as an open problem. We settle this problem, that is, resolve the nature of this polynomial “jump” of the clique chromatic number of the random graph around edge‐probability . Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of up to logarithmic factors for any edge‐probability .

Tight Bounds on the Clique Chromatic Number

The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.

Lower bounds on the clique-chromatic numbers of some distance graphs

Lower bounds on the clique-chromatic numbers of some distance graphs

New bounds on clique-chromatic numbers of Johnson graphs

The clique-chromatic number χc(G) of graph G is defined as the minimum number k such that there exists a colouring of the vertices of G in k colours that satisfies the following property: all maximal cliques in G (except for the isolated vertices) contain vertices of at least two colours. In this paper we significantly improve lower bounds on this value for some families of Johnson graphs. We also obtain a new upper bound on clique-chromatic number for G(n,r,r−1) and G(n,3,1). Finally, we provide the exact value of clique-chromatic number for G(n,2,1).

New Bounds for the Clique-Chromatic Numbers of Johnson Graphs

We significantly improve lower bounds on the clique-chromatic numbers for some families of Johnson graphs. A new upper bound on the clique-chromatic numbers for G(n, r, r – 1) and G(n, 3, 1) is obtained. Finally, the exact value of the clique-chromatic number for G(n, 2, 1) is provided.

Clique-Coloring of $$K_{3,3}$$-Minor Free Graphs

A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this paper, we generalize these results to $$K_{3,3}$$ -minor free ( $$K_{3,3}$$ -subdivision free) graphs.

Clique Chromatic Numbers of Intersection Graphs

Clique Chromatic Numbers of Intersection Graphs

Clique Chromatic Numbers of Intersection Graphs

Clique Chromatic Numbers of Intersection Graphs

Clique Colourings of Geometric Graphs

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.

Clique coloring of binomial random graphs

A clique coloring of a graph is a coloring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colors in such a coloring is the clique chromatic number. In this paper, we study the asymptotic behavior of the clique chromatic number of the random graph 𝒢(n,p) for a wide range of edge‐probabilities p = p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.

Clique coloring of dense random graphs

AbstractThe clique chromatic number of a graph is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random graph is, with high probability, . This settles a problem of McDiarmid, Mitsche, and Prałat who proved that it is with high probability.

Coloring the cliques of line graphs

The weak chromatic number, or clique chromatic number (CCHN) of a graph is the minimum number of colors in a vertex coloring, such that every maximal clique gets at least two colors. The weak chromatic index, or clique chromatic index (CCHI) of a graph is the CCHN of its line graph.Most of the results here are upper bounds for the CCHI, as functions of some other graph parameters, and contrasting with lower bounds in some cases. Algorithmic aspects are also discussed; the main result within this scope (and in the paper) shows that testing whether the CCHI of a graph equals 2 is NP-complete. We deal with the CCHN of the graph itself as well.

Efficient Algorithms for Clique-Colouring and Biclique-Colouring Unichord-Free Graphs

The class of unichord-free graphs was recently investigated in the context of vertex-colouring (Trotignon and Vuskovic in J Graph Theory 63(1): 31---67, 2010), edge-colouring (Machado et al. in Theor Comput Sci 411(7---9): 1221---1234, 2010) and total-colouring (Machado and de Figueiredo in Discrete Appl Math 159(16): 1851---1864, 2011). Unichord-free graphs proved to have a rich structure that can be used to obtain interesting results with respect to the study of the complexity of colouring problems. In particular, several surprising complexity dichotomies of colouring problems are found in subclasses of unichord-free graphs. In the present work, we investigate clique-colouring and biclique-colouring problems restricted to unichord-free graphs. We show that the clique-chromatic number of a unichord-free graph is at most 3, and that the 2-clique-colourable unichord-free graphs are precisely those that are perfect. Moreover, we describe an O(nm)-time algorithm that returns an optimal clique-colouring of a unichord-free graph input. We prove that the biclique-chromatic number of a unichord-free graph is either equal to or one greater than the size of a largest twin set. Moreover, we describe an $$O(n^2m)$$O(n2m)-time algorithm that returns an optimal biclique-colouring of a unichord-free graph input. The clique-chromatic and the biclique-chromatic numbers are not monotone with respect to induced subgraphs. The biclique-chromatic number presents an extra unexpected difficulty, as it is not the maximum over the biconnected components, which we overcome by considering additionally the star-biclique-chromatic number.

Graphs with large clique-chromatic numbers

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.

Perfect graphs of arbitrarily large clique-chromatic number

We prove that there exist perfect graphs of arbitrarily large clique-chromatic number. These graphs can be obtained from cobipartite graphs by repeatedly gluing along cliques. This negatively answers a question raised by Duffus et al. (1991) [5].

Coloring clique-hypergraphs of graphs with no subdivision of [formula omitted

A clique-coloring of a graph G is a coloring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, H(G), of a graph G has V(G) as its set of vertices and the maximal cliques of G as its hyperedges. A (vertex) coloring of H(G) with no monochromatic hyperedge is a clique-coloring of G. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. Every planar graph has been proved to be 3-clique-colorable and every claw-free planar graph, different from an odd cycle, has been proved to be 2-clique-colorable. In this paper we first generalize the result of planar graphs to K5-minor-free graphs. Furthermore, we generalize the result of claw-free planar graphs to K5-subdivision-free graphs and give a polynomial-time algorithm to find a 2-clique-coloring of K5-subdivision-free graphs.

Restricted coloring problems on Graphs with few P 4’s

In this paper, we obtain linear time algorithms to determine the acyclic chromatic number, the star chromatic number, the non repetitive chromatic number and the clique chromatic number of P 4-tidy graphs and (q, q − 4)-graphs, for every fixed q, which are the graphs such that every set with at most q vertices induces at most q − 4 distinct P 4’s. These classes include cographs and P 4-sparse graphs. We also obtain a linear time algorithm to compute the harmonious chromatic number of connected P 4-tidy graphs and connected (q, q − 4)-graphs. All these coloring problems are known to be NP-hard for general graphs. These algorithms are fixed parameter tractable on the parameter q(G), which is the minimum q such that G is a (q, q − 4)-graph. We also prove that every connected (q, q − 4)-graph with at least q vertices is 2-clique-colorable and that every acyclic coloring of a cograph is also nonrepetitive, generalizing the main result of Lyons (2011).

Colouring Clique-Hypergraphs of Circulant Graphs

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, $${\mathcal{H}(G)}$$ , of a graph G has V(G) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of $${\mathcal{H}(G)}$$ is a clique-colouring of G. Determining the clique-chromatic number, the least number of colours for which a graph G admits a clique-colouring, is known to be NP-hard. In this work, we establish that the clique-chromatic number of powers of cycles is equal to two, except for odd cycles of size at least five, that need three colours. For odd-seq circulant graphs, we show that their clique-chromatic number is at most four, and determine the cases when it is equal to two. Similar bounds for the chromatic number of these graphs are also obtained.

On clique-colouring of graphs with few P4’s

Abstract Let G=(V,E) be a graph with n vertices. A clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured. A k-clique-colouring is a clique-colouring that uses k colours. The clique-chromatic number of a graph G is the minimum k such that G has a k-clique-colouring. In this paper we will use the primeval decomposition technique to find the clique-chromatic number and the clique-colouring of well known classes of graphs that in some local sense contain few P 4’s. In particular we shall consider the classes of extended P 4-laden graphs, p-trees (graphs which contain exactly n−3 P 4’s) and (q,q−3)-graphs, q≥7, such that no set of at most q vertices induces more that q−3 distincts P 4’s. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cographs, P 4-reducible graphs, P 4-sparse graphs, extended P 4-reducible graphs, extended P 4-sparse graphs, P 4-extendible graphs, P 4-lite graphs, P 4-tidy graphs and P 4-laden graphs that are included in the class of extended P 4-laden graphs.

Colouring clique-hypergraphs of circulant graphs

Abstract A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, H ( G ) , of a graph G has V ( G ) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of H ( G ) is a clique-colouring of G. Determining the clique-chromatic number, the least number for which a graph G admits a clique-colouring, is known to be NP-hard. We establish that the clique-chromatic number for powers of cycles is equal to two, except for odd cycles of size at least five, that need three colours. For odd-seq circulant graphs, we show that their clique-chromatic number is at most four, and determine the cases when it is equal to two.