On the Difference Between the Chromatic and Cochromatic Number

Chromatic Number Versus Cochromatic Number in Graphs with Bounded Clique Number

Partitioning graphs into complete and empty graphs

The cochromatic number of a graph G, denoted by z(G), is the minimum number of subsets into which the vertex set of G can be partitioned so that each sbuset induces an empty or a complete subgraph of G. In this paper we introduce the problem of determining for a surface S, z(S), which is the maximum cochromatic number among all graphs G that embed in S. Some general bounds are obtained; for example, it is shown that if S is orientable of genus at least one, or if S is nonorientable of genus at least four, then z(S) is nonorientable of genus at least four, then z(S)≤χ(S). Here χ(S) denotes the chromatic number S. Exact results are obtained for the sphere, the Klein bottle, and for S. It is conjectured that z(S) is equal to the maximum n for which the graph Gn = K1 ∪ K2 ∪ … ∪ Kn embeds in S.

The cochromatic number of a graph G = (V, E) is the smallest number of parts in a partition of V in which each part is either an independent set or induces a complete subgraph. We show that if the chromatic number of G is n, then G contains a subgraph with cochromatic number at least . This is tight, up to the constant factor, and settles a problem of Erdös and Gimbel. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 295–297, 1997

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs. When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l. In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l). 1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l. 2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0. 3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.

The cochromatic number Z(G) of a graph G is the fewest number of colors needed to color the vertices of G so that each color class is a clique or an independent set. In a fractional cocoloring of G a non-negative weight is assigned to each clique and independent set so that for each vertex v, the sum of the weights of all cliques and independent sets containing v is at least one. The smallest total weight of such a fractional cocoloring of G is the fractional cochromatic number \(Z_f(G)\). In this paper we prove results for the fractional cochromatic number \(Z_f(G)\) that parallel results for Z(G) and the well studied fractional chromatic number \(\chi _f{(G)}\). For example \(Z_f(G)=\chi _f(G)\) when G is triangle-free, except when the only nontrivial component of G is a star. More generally, if G contains no k-clique, then \(Z_f(G)\le \chi _f(G)\le Z_f(G)+R(k,k)\), where R(k, k) is the minimum integer n such that every n-vertex graph has a k-clique or an independent set of size k. Moreover, every graph G with \(\chi _f(G)=m\) contains a subgraph H with \(Z_f(H)\ge (\frac{1}{4} - o(1))\frac{m}{\log _2 m}\). We also prove that the maximum value of \(Z_f(G)\) over all graphs G of order n is \(\varTheta (n/\log n)\), and the maximum over all graphs embedded on an orientable surface of genus g is \(\varTheta (\sqrt{g} / \log g)\).

Partitioning extended [formula omitted]-laden graphs into cliques and stable sets

Fixed-parameter algorithms for the cocoloring problem