Lovász and Cherkassky discovered in the 1970s independently that if G is a finite graph with given set T of terminal vertices such inner Eulerian respect to T, then maximal number edge-disjoint paths connecting distinct ∑t∈Tλ(t,T−t) where λ local edge-connectivity function. The optimality system T-paths Lovász-Cherkassky theorem witnessed by existence certain cuts Menger's theorem. infinite gen...

A path (resp. cycle) decomposition of a graph G is a set of edge-disjoint paths (resp. cycles) of G that covers the edge set of G. Gallai (1966) conjectured that every graph on n vertices admits a path decomposition of size at most ⌊(n+ 1)/2⌋, and Hajós (1968) conjectured that every Eulerian graph on n vertices admits a cycle decomposition of size at most ⌊(n−1)/2⌋. Gallai’s Conjecture was veri...

A graph G is Nm-locally connected if for every vertex v in G, the vertices not equal to v and with distance at most m to v induce a connected subgraph in G. We show that both connected N2-locally connected claw-free graph and 3-edge-connected N3-locally connected claw-free graph have connected even [2, 4]-factors, which settle a conjecture by Li in [6].

For a graph G = (V,E) and a vertex v ∈ V , let T (v) be a local trace at v, i.e. T (v) is an Eulerian subgraph of G such that every walk W (v), with start vertex v can be extended to an Eulerian tour in T (v). We prove that every maximum edge-disjoint cycle packing Z∗ of G induces a maximum trace T (v) at v for every v ∈ V . Moreover, if G is Eulerian then sufficient conditions are given that g...

We show that the edge disjoint paths problem is NP-complete in directed or undirected rectangular grids, even if the union G + H of the supply and the demand graph is Eulerian.

Let {Gp1, Gp2, . . .} be an infinite sequence of graphs with Gpn having pn vertices. This sequence is called Kp-removable if Gp1 ∼= Kp, and Gpn − S ∼= Gp(n−1) for every n ≥ 2 and every vertex subset S of Gpn that induces a Kp. Each graph in such a sequence has a high degree of symmetry: every way of removing the vertices of any fixed number of disjoint Kp’s yields the same subgraph. Here we con...

The odd edge connectivity of a graph G, denoted by o(G), is the size of a smallest odd edge cut of the graph. Let S be any given surface and be a positive real number. We proved that there is a function fS( ) (depends on the surface S and lim !0 fS( )1⁄41) such that any graph G embedded in S with the odd-edge connectivity at least fS( ) admits a nowhere-zero circular (2þ )-flow. Another major r...

We initiate the study of enumerating linear subspaces alternating matrices over finite fields with explicit coordinates. present q-analogues Gilbert's formula for connected graphs (Gilbert (1956) [5]), and Read's c-coloured (Read (1960) [14]). also develop an analogue Riddell's relating exponential generating function that (Riddell's (1951) [15]), building on Eulerian functions developed by Sri...

The study of genome rearrangement has many flavours, but they all are somehow tied to edit distances on variations of a multi-graph called the breakpoint graph. We study a weighted 2-break distance on Eulerian 2-edge-colored multi-graphs, which generalizes weighted versions of several Double Cut and Join problems, including those on genomes with unequal gene content. We affirm the connection be...

Fleischner, H., G. Sabidussi and E. Wenger, Transforming eulerian trails, Discrete Mathematics 109 (1992) 103-116. In this paper a set of transformations (K-transformations) between eulerian trails is investigated. It is known that two arbitrary eulerian trails can be transformed into each other by a sequence of K-transformations. For compatible eulerran trails the set of K-transformations is a...