Edge-disjoint paths in planar graphs

Given a planar graph G = (V, E), find k edge-disjoint paths in G connecting k pairs of terminals specified on the outer face of G. Generalizing earlier results of Okamura and Seymour (J. Combin. Theory Ser. B31 (1981), 75–81) and of the author (Combinatorica2, No. 4 (1982), 361–371), we solve this problem when each node of G not on the outer face has even degree. The solution involves a good characterization for the solvability and the proof gives rise to an algorithm of complexity O(|V|³log|V|). In particular, the integral multicommodity flow problem is proved to belong to the problem class P when the underlying graph is outerplanar.     

Edge-disjoint cycles in regular directed graphs

We prove that any k-regular directed graph with no parallel edges contains a collection of at least O(k²) edge-disjoint cycles; we conjecture that in fact any such graph contains a collection of at least (^k+1₂) disjoint cycles, and note that this holds for k ≤ 3. © 1996 John Wiley & Sons, Inc.     

赋权图中的路和圈

本文研究了赋权图中的最长路和最长圈,将关于非赋权图中最长路和最长圈的一些结果推广到赋权图上.     

Long paths and cycles in oriented graphs

We obtain several sufficient conditions on the degrees of an oriented graph for the existence of long paths and cycles. As corollaries of our results we deduce that a regular tournament contains an edge-disjoint Hamilton cycle and path, and that a regular bipartite tournament is hamiltonian.     

Long paths and cycles in tough graphs

For a graphG, letp(G) andc(G) denote the length of a longest path and cycle, respectively. Let (t,n) be the minimum ofp(G), whereG ranges over allt-tough connected graphs onn vertices. Similarly, let (t,n) be the minimum ofc(G), whereG ranges over allt-tough 2-connected graphs onn vertices. It is shown that for fixedt>0 there exist constantsA, B such that (t,n)A·log(n) and (t,n)·log((t,n))B·log(n). Examples are presented showing that fort1 there exist constantsA, B such that (t,n)A·log(n) and (t,n)B· log(n). It is conjectured that (t,n) B·log(n) for some constantB. This conjecture is shown to be valid within the class of 3-connected graphs and, as conjectured in Bondy [1] forl=3, within the class of 2-connectedK _1.l-free graphs, wherel is fixed.     

Long cycles and paths in distance graphs

For n∈N and D⊆N, the distance graph has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant c_D such that the greatest common divisor of the integers in D is 1 if and only if for every n, has a component of order at least n−c_D if and only if for every n≥c_D+3, has a cycle of order at least n−c_D. Furthermore, we discuss some consequences and variants of this result.     

Long cycles in graphs without hamiltonian paths

For a graph G, p(G) and c(G) denote the order of a longest path and a longest cycle of G, respectively. Bondy and Locke [J.A. Bondy, S.C. Locke, Relative length of paths and cycles in 3-connected graphs, Discrete Math. 33 (1981) 111-122] consider the gap between p(G) and c(G) in 3-connected graphs G. Starting with this result, there are many results appeared in this context, see [H. Enomoto, J. van den Heuvel, A. Kaneko, A. Saito, Relative length of long paths and cycles in graphs with large degree sums, J. Graph Theory 20 (1995) 213-225; M. Lu, H. Liu, F. Tian, Relative length of longest paths and cycles in graphs, Graphs Combin. 23 (2007) 433-443; K. Ozeki, M. Tsugaki, T. Yamashita, On relative length of longest paths and cycles, preprint; I. Schiermeyer, M. Tewes, Longest paths and longest cycles in graphs with large degree sums, Graphs Combin. 18 (2002) 633-643]. In this paper, we investigate graphs G with p(G)−c(G) at most 1 or at most 2, but with no hamiltonian paths. Let G be a 2-connected graph of order n, which has no hamiltonian paths. We show two results as follows: (i) if , then p(G)−c(G)≤1, and (ii) if σ₄(G)≥n+3, then p(G)−c(G)≤2.     

Long paths and large cycles in finite graphs

Ore-type sufficient conditions ensuring the existence of a large cycle passing through any given path of length s for (s + 2)-connected graphs are given, and the extremal cases are characterized.     

Long induced paths and cycles in Kneser graphs

We present some lower and upper bounds on the length of the maximum induced paths and cycles in Kneser graphs.     

Parity theorems for paths and cycles in graphs

We extend an elegant proof technique of A. G. Thomason, and deduce several parity theorems for paths and cycles in graphs. For example, a graph in which each vertex is of even degree has an even number of paths if and only if it is of even order, and a graph in which each vertex is of odd degree has an even number of paths if and only if its order is a multiple of four. Our results have implications for generalized friendship graphs and their conjectured nonexistence.     

Longest paths and cycles in bipartite oriented graphs

In this paper we obtain two sufficient conditions, Ore type (Theorem 1) and Dirac type (Theorem 2), on the degrees of a bipartite oriented graph for ensuring the existence of long paths and cycles. These conditions are shown to be the best possible in a sense.     

Forbidden paths and cycles in ordered graphs and matrices

Assume % MathType!End!2!1! and let Ω⊂R ^N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.     

Hamilton paths and cycles in vertex-transitive graphs of order

It is shown that every connected vertex-transitive graph of order 6p, where p is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order 6p which is not genuinely imprimitive contains a Hamilton cycle.     

Longest paths and cycles in K1,3-free graphs

In this article we show that the standard results concerning longest paths and cycles in graphs can be improved for K_1,3-free graphs. We obtain as a consequence of these results conditions for the existence of a hamiltonian path and cycle in K_1,3-free graphs.     

Infinite paths in planar graphs IV,dividing cycles

Nash‐Williams conjectured that a 4‐connected infinite planar graph contains a spanning 2‐way infinite path if, and only if, the deletion of any finite set of vertices results in at most two infinite components. In this article, we prove this conjecture for graphs with no dividing cycles and for graphs with infinitely many vertex disjoint dividing cycles. A cycle in an infinite plane graph is called dividing if both regions of the plane bounded by this cycle contain infinitely many vertices of the graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 173–195, 2006