A graph is H-free if it has no induced subgraph isomorphic to H. Let F(D; H) be the family of H-free connected graphs with maximum degree at most D. Let ex∗(D; H) = max{|E(G)| : G ∈ F(D; H)}; this is finite only when H is a disjoint union of paths. In that case, ex∗(D; H) ∈ O(D2 ex∗(D; Pm)), where Pm is the longest path in H. For m ≥ 6, we determine ex∗(D; Pm) asymptotically (with m fixed), sho...

We investigate graphs with the property that all longest paths or all longest cycles have empty intersection. In this paper, we find such graphs as subgraphs of cubic lattices.

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann conjectured that l ≤ 2c− 1, where l (c, respectively) is the number of vertices in a longest path (longest cycle) of a strong semicomplete multipartite digraph. The bound on l is sharp. We s...

The detour order of a graph G, denoted by τ (G) , is the order of a longest path in G. A partition of the vertex set of G into two sets, A and B, such that τ(〈A〉) ≤ a and τ(〈B〉) ≤ b is called an (a, b)-partition of G. If G has an (a, b)-partition for every pair (a, b) of positive integers such that a + b = τ(G), then we say that G is τ -partitionable. The Path Partition Conjecture (PPC), which ...

1 Weighted automata 1 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Shortest and longest paths . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Optimal mean payoff . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Weighted temporal logics . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Energy constraints . . . . . . . . . . . . . . . . . . . . . . ...

We consider two problems that arise in designing two-level star networks taking into account service quality considerations. Given a set of nodes with pairwise traffic demand and a central hub, we select p hubs and connect them to the central hub with direct links and then we connect each nonhub node to a hub. This results in a star/star network. In the first problem, called the Star p-hub Cent...

st-orientations (st-numberings) or bipolar orientations of undirected graphs are central to many graph algorithms and applications. Several algorithms have been proposed in the past to compute an st-orientation of a biconnected graph. In this paper, we present new algorithms that compute such orientations with certain (parameterized) characteristics in the final st-oriented graph, such as the l...