Cochromatic Number Research Articles

Some topics in cochromatic theory

GivenG, a graph, the cochromatic number,Z(G), ofG is the fewest number of sets into which the vertex set can be partitioned so that each set induces a complete or an empty graph. A graph is critically cochromatic if the removal of any of its vertices decreases its cochromatic number. A graph is uniquely cochromatic if there is exactly one partition of minimum order in which each set induces a complete or an empty graph. A graph is comaximal if the removal of any edge increases its cochromatic number. These and related concepts are examined.

Uniquely cocolourable graphs

A graphG is uniquelyn-cocolourable if (its cochromatic number)z(G) = n and alln-cocolourings ofG induce the same partition of its vertex set. We prove that there are infinitely many uniquelyn-cocolourable graphs and that the numbers of complete and empty colour classes can be prescribed. It is also shown that everyn-cocolourable graph is an induced subgraph of a uniquelyn-cocolourable graph.

Note on the cochromatic number of several surfaces

AbstractThe 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 subset induces an empty or a complete subgraph of G. In an earlier work, the author considered the problem of determining z(S), the maximum cochromatic number among all graphs that embed in a surface S. The value of z(S) was found for the sphere, the Klein bottle, and for the nonorientable surface of genus 4. In this note, some recent results of Albertson and Hutchinson are used to determine the cochromatic numbers of the projective plane and the nonorientable surface of genus 3. These results lend further evidence to support the conjecture that z(S) is equal to the maximum n for which the graph Gn = K1 U K2 U … U Kn embeds in S.

Cochromatic Number and the Genus of a Graph

AbstractThe 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.