Pósa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgraph of G whose components are nontrivial paths. Let k be the numbe...

A semicomplete multipartite or semicomplete c-partite digraph D is a biorientation of a c-partite graph. A semicomplete multipartite digraph D is called strongly quasiHamiltonian-connected, if for any two distinct vertices x and y of D, there is a path P from x to y such that P contains at least one vertex from each partite set of D. In this paper we show that every 4-strong semicomplete multip...

Transversals in r-partite graphs with various properties are known to have many applications in graph theory and theoretical computer science. We investigate f -bounded transversals (or f -BT), that is, transversals whose connected components have order at most f . In some sense we search for the the sparsest f -BT-free graphs. We obtain estimates on the smallest maximum degree that 3-partite a...

MOTIVATION We consider the problem of multiple alignment of protein sequences with the goal of achieving a large SP (Sum-of-Pairs) score. RESULTS We introduce a new graph-based method. We name our method QOMA (Quasi-Optimal Multiple Alignment). QOMA starts with an initial alignment. It represents this alignment using a K-partite graph. It then improves the SP score of the initial alignment th...

Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑ u∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it c...

Abstract Generalizing well‐known results of Erdős and Lovász, we show that every graph contains a spanning ‐partite subgraph with , where is the edge‐connectivity . In particular, together result due to Nash‐Williams Tutte, this implies 7‐edge‐connected bipartite whose edge set decomposes into two edge‐disjoint trees. We best possible as it does not hold for infinitely many 6‐edge‐connected gra...

We look at structures that must be removed (or reversed) in order to make acyclic a given oriented graph. For directed graph $H$ and an $G$, let $f_H(G)$ the maximum number of pairwise disjoint copies can found {\em all} feedback arc sets $G$. In particular, $G$ acyclic, one remove reverse) $H$. Most intriguing is case where $k$-clique, parameter denoted by $f_k(G)$. Determining $f_k(G)$ for ar...

The chromatic capacity χcap(G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f :N → N such that χcap(G) ≥ f(χ(G)) for almost every graph G, where χ denotes the chromatic number. We show tha...

We find the minimal number of links in an embedding of any complete k-partite graph on 7 vertices (including K 7 , which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete k-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with min...