Cycles, Disjoint Spanning Trees and Orientations of Graphs
Yanting Liang · 2012
A graph G is hamiltonian-connected if any two of its vertices are connected by a Hamilton path (a path including every vertex of G); and G is s-hamiltonian-connected if the deletion of any vertex subset with at most s vertices results in a hamiltonian-connected graph. We prove that the line graph of a (t + 4)-edge-connected graph is (t + 2)-hamiltonian-connected if and only if it is (t + 5)-connected, and for s ≥ 2 every (s + 5)-connected line graph is s-hamiltonian-connected.;For integers l and k with l > 0, and k ≥ 0, Ch( l, k) denotes the collection of h-edge-connected simple graphs G on n vertices such that for every edge-cut X with 2 ≤ |X| ≤ 3, each component of G -- X has at least (n -- k)/l vertices. We prove that for any integer k > 0, there exists an integer N = N( k) such that for any n ≥ N, any graph G ∈ C2(6, k) on n vertices is supereulerian if and only if G cannot be contracted to a member in a well characterized family of graphs.;An orientation of an undirected graph G is a mod (2 p + 1)-orientation if under this orientation, the net out-degree at every vertex is congruence to zero mod 2p + 1. A graph H is mod (2p + 1)-contractible if for any graph G that contains H as a subgraph, the contraction G/H has a mod (2p + 1)-orientation if and only if G has a mod (2p + 1)-orientation (thus every mod (2p + 1)-contractible graph has a mod (2p + 1)-orientation). Jaeger in 1984 conjectured that every (4p)-edge-connected graph has a mod (2p + 1)-orientation. It has also been conjectured that every (4p + 1)-edge-connected graph is mod (2 p + 1)-contractible. We investigate graphs that are mod (2 p + 1)-contractible, and as applications, we prove that a complete graph Km is (2p + 1)-contractible if and only if m ≥ 4p + 1; that every (4p -- 1)-edge-connected K4-minor free graph is mod (2p + 1)-contractible, which is best possible in the sense that there are infinitely many (4p -- 2)-edge-connected K4-minor free graphs that are not mod (2p + 1)-contractible; and that every (4p)-connected chordal graph is mod (2p + 1)-contractible. We also prove that the above conjectures on line graphs would imply the truth of the conjectures