We study the class N of graphs, the right-angled Artin groups defined on which do not contain surface subgroups. We prove that a presumably smaller class N ′ is closed under amalgamating along complete subgraphs, and also under adding bisimplicial edges. It follows that chordal graphs and chordal bipartite graphs belong to N ′.

Let H be a graph on n vertices and G a collection of n subgraphs of H, one for each vertex, where G is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of G and any two members share an edge whenever the corresponding vertices are adjacent in H and share no edges whenever the corresponding vertices are nonadjacent in H. In this paper, we are concerned with ...

The existence of an on-line competitive algorithm for coloring bipartite graphs remains a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs, in particular for P7-free bipartite graphs. We propose a new on-line competitive coloring algorithm for P8-free bipartite graphs. Our proof technique impro...

Let H = {A1, . . . , An, B1, . . . , Bn} be a collection of 2n subgraphs of the complete bipartite graph Kn,n. The collection H is called an orthogonal double cover (ODC) of Kn,n if each edge of Kn,n occurs in exactly two of the graphs in H; E(Ai) ∩ E(Aj) = φ = E(Bi) ∩ E(Bj) for every i, j ∈ {1, . . . , n} with i = j, and for any i, j ∈ {1, . . . , n}, |E(Ai)∩E(Bj)| = 1. If Ai ∼= G ∼= Bi for al...

A graph is connected-homogeneous if any isomorphism between finite connected induced subgraphs extends to an automorphism of the graph. In this paper we classify the countably infinite connectedhomogeneous graphs. We prove that if Γ is connected countably infinite and connected-homogeneous then Γ is isomorphic to one of: Lachlan and Woodrow’s ultrahomogeneous graphs; the generic bipartite graph...

In this article we consider graph partitions with prescribed patterns. Specifically a 2-colouring of the vertex set such that one colour induces an independent and other complete bipartite graph. We determine list obstructions (minimal forbidden induced subgraphs) for chordal graphs. This result completes knowledge on minimal 3 × matrix partition problems

The concepts of covering and matching in fuzzy graphs using strong arcs are introduced and obtained the relationship between them analogous to Gallai’s results in graphs. The notion of paired domination in fuzzy graphs using strong arcs is also studied. The strong paired domination number γspr of complete fuzzy graph and complete bipartite fuzzy graph is determined and obtained bounds for the s...