Spanningk-ended trees of bipartite graphs

Spanning $k$-trees of Bipartite Graphs

A tree is called a k-tree if its maximum degree is at most k. We prove the following theorem. Let k ⩾ 2 be an integer, and G be a connected bipartite graph with bipartition (A,B) such that |A| ⩽ |B| ⩽ (k − 1)|A| + 1. If σk(G) ⩾ |B|, then G has a spanning k-tree, where σk(G) denotes the minimum degree sum of k independent vertices of G. Moreover, the condition on σk(G) is sharp. It was shown by ...

Bipartite graphs with even spanning trees

Steiner trees in uniformly quasi-bipartite graphs

The area of approximation algorithms for the Steiner tree problem in graphs has seen continuous progress over the last years. Currently the best approximation algorithm has a performance ratio of 1.550. This is still far away from 1.0074, the largest known lower bound on the achievable performance ratio. As all instances resulting from known lower bound reductions are uniformly quasi-bipartite,...

Contracting chordal graphs and bipartite graphs to paths and trees

We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems, which we call Tree Contraction and Path Contraction, respectively, are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n + m)...

Interval colorings of complete bipartite graphs and trees

In the work interval colorings [1] of complete bipartite graphs and trees are investigated. The obtained results were announced in [2]. Non defined concepts can be found in [3, 4]. Let G = (V (G), E(G)) be an undirected graph without multiple edges and loops. The degree of a vertex x in G is denoted by dG(x), the greatest degree of vertices – by ∆(G), the chromatic index of G – by χ(G). Interva...