Integer flows and cycle covers

AbstractResults related to integer flows and cycle covers are presented. A cycle cover of a graph G is a collection C of cycles of G which covers all edges of G; C is called a cycle m-cover of G if each edge of G is covered exactly m times by the members of C. By using Seymour's nowhere-zero 6-flow theorem, we prove that every bridgeless graph has a cycle 6-cover associated to covering of the edges by 10 even subgraphs (an even graph is one in which each vertex is of even degree). This result together with the cycle 4-cover theorem implies that every bridgeless graph has a cycle m-cover for any even number m ≥ 4. We also prove that every graph with a nowhere-zero 4-flow has a cycle cover C such that the sum of lengths of the cycles in C is at most |E(G)| + |V(G)| − 2, unless G belongs to a very special class of graphs

Hamiltonian properties of graphs with large neighborhood unions

AbstractLet G be a graph of order n, σk = min{ϵi=1kd(νi): {ν1,…, νk} is an independent set of vertices in G}, NC = min{|N(u)∪ N(ν)|: uν∉E(G)} and NC2 = min{|N(u)∪N(ν)|: d(u,ν)=2}. Ore proved that G is hamiltonian if σ2⩾n⩾3, while Faudree et al. proved that G is hamiltonian if G is 2-connected and NC⩾13(2n−1). It is shown that both results are generalized by a recent result of Bauer et al. Various other existing results in hamiltonian graph theory involving degree-sums or cardinalities of neighborhood unions are also compared in terms of generality. Furthermore, some new results are proved. In particular, it is shown that the bound 13(2n−1) on NC in the result of Faudree et al. can be lowered to 13(2n−1), which is best possible. Also, G is shown to have a cycle of length at least min{n, 2(NC2)} if G is 2-connected and σ3⩾n+2. A Dλ-cycle (Dλ-path) of G is a cycle (path) C such that every component of G−V(C) has order smaller than λ. Sufficient conditions of Lindquester for the existence of Hamilton cycles and paths involving NC2 are extended to Dλ-cycles and Dλ-paths

Anomalous Cooper pair interference on Bi2Te3 surface

It is believed that the edges of a chiral p-wave superconductor host Majorana modes, relating to a mysterious type of fermions predicted seven decades ago. Much attention has been paid to search for p-wave superconductivity in solid-state systems, including recently those with strong spin-orbit coupling (SOC). However, smoking-gun experiments are still awaited. In this work, we have performed phase-sensitive measurements on particularly designed superconducting quantum interference devices constructing on the surface of topological insulators Bi2Te3, in such a way that a substantial portion of the interference loop is built on the proximity-effect-induced superconducting surface. Two types of Cooper interference patterns have been recognized at low temperatures. One is s-wave like and is contributed by a zero-phase loop inhabited in the bulk of Bi2Te3. The other, being identified to relate to the surface states, is anomalous for that there is a phase shift between the positive and negative bias current directions. The results support that the Cooper pairs on the surface of Bi2Te3 have a 2\pi Berry phase which makes the superconductivity p_x+ip_y-wave-like. Mesoscopic hybrid rings as constructed in this experiment are presumably arbitrary-phase loops good for studying topological quantum phenomena.Comment: supplementary material adde

Covering weighted graphs by even subgraphs

AbstractA weighted graph is one in which each edge e is assigned a nonnegative number w(e), called the weight of e. The weight of a subgraph is the sum of the weights of its edges. An even graph is a graph every vertex of which is of even degree. A cover of a graph G is a collection of its subgraphs which together cover each edge of G at least once. A cover is called an (l, m)-cover if each edge of G is covered either exactly l or exactly m times. We prove that every bridgeless graph has a (2, 4)-cover by four even subgraphs of total weight at most (209) w(G). As a corollary, this result yields a weighted generalization of a result found by J. C. Bermond, B. Jackson, and F. Jaeger (J. Combin. Theory Ser. B 35, 1983, 299–308) and N. Alon and M. Tarsi (SIAM J. Algebraic Discrete Methods 6, 1985, 345–350)

Cycles in 4-Connected Planar Graphs

Let G be a 4-connected planar graph on n vertices. Previous results show that G contains a cycle of length k for each k ∈ {n, n − 1, n − 2, n − 3} with k ≥ 3. These results are proved using the “Tutte path” technique, and this technique alone cannot be used to obtain further results in this direction. One approach to obtain further results is to combine Tutte paths and contractible edges. In this paper, we demonstrate this approach by showing that G also has a cycle of length k for each k ∈ {n − 4, n − 5, n − 6} with k ≥ 3. This work was partially motivated by an old conjecture of Malkevitch

Cycles in 2-connected graphs

Let Gn be a class of graphs on n vertices.For an integer c, let ex(Gn,c) be the smallest integer such that if G is a graph in Gn with more than ex(Gn,c) edges, then G contains a cycle of length more than c.A classical result of Erdös and Gallai is that if Gn is the class of all simple graphs on n vertices, then ex(Gn,c) = c 2 (n − 1).The result is best possible when n − 1 is divisible by c − 1, in view of the graph consisting of copies of Kc all having exactly one vertex in common.Woodall improved the result by giving best possible bounds for the remaining cases when n − 1 is not divisible by c − 1, and conjectured that if Gn is the class of all 2-connected simple graphs on n vertices, then ex(Gn,c) = max{f (n, 2, c), f (n, ⌊c/2⌋,c)}, c+1−t where f (n, t, c) = 2 + t(n − c − 1 + t),2�t �c/2, is the number of edges in the graph obtained from Kc+1−t by adding n − (c + 1 − t) isolated vertices each joined to the same t vertices of Kc+1−t.By using a result of Woodall together with an edge-switching technique, we confirm Woodall’s conjecture in this paper