In this paper, we address the minimum 2-edge connected spanning subgraph problem and the graph-TSP in regular bipartite graphs. For these problems, we present new approximation algorithms, each of which finds a restricted 2-factor close to a Hamilton cycle in the first step. We first prove that every regular bipartite graph of degree at least three has a square-free 2-factor. This immediately l...

We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edgeweighted undirected planar graphs that are resistant to edge or vertex removal. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-w...

Those (2m−1)-edge-colourings of a spanning subgraph of K2m , consisting of Kr and independent edges, that can be embedded in a (2m− 1)-edge-colouring of K2m are characterised.

Let G be a 2-connected claw-free graph. We show that • if G is N1,1,4-free or N1,2,2-free or Z5-free or P8-free, respectively, then G has a spanning eulerian subgraph (i.e. a spanning connected even subgraph) or its closure is the line graph of a graph in a family of well-defined graphs, • if the minimum degree δ(G) ≥ 3 and G is N2,2,5-free or Z9-free, respectively, then G has a spanning euleri...

Mader (1985) proved that every strongly $k$-connected $n$-vertex digraph contains a strongly $k$-connected spanning subgraph with at most $2kn - 2k^2$ edges, and this is sharp. For dense strongly $k$-connected digraphs, the bound can be significantly improved for dense strongly $k$-connected digraphs. Let $\overline{\Delta}(D)$ be the maximum degree of the complement of the underlying undirecte...

Let G be a simple balanced bipartite graph on 2n vertices, δ = δ(G)/n, and ρ0 = δ+ √ 2δ−1 2 . If δ ≥ 1/2 then G has a ⌊ρ0n⌋-regular spanning subgraph. The statement is nearly tight.

Let G be a finite graph and A be a subgraph of Aut(G). We give a necessary and sufficient condition for the graph G to have an A-invariant spanning tree.

We give sufficient conditions for a tripartite graph to have a spanning subgraph consisting of complete graphs of order 3. This is applied to bound the strong chromatic number of a graph.

Hoffmann–Ostenhof’s conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, matching and 2-regular subgraph. In this paper, we show holds for claw-free subcubic graphs 4-chordal graphs.

We investigate the MinimumWeight 2-Edge-Connected Spanning Subgraph (2-ECSS) problem in an arbitrary metric space of doubling dimension and show a polynomial time randomized (1+ǫ)-approximation algorithm.