6,475 research outputs found
Long cycles in graphs containing a 2-factor with many odd components
We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+d(c)+d(w)n+2 for every tiple u, v, w of independent vertices. As a corollary we obtain the follwing improvement of a conjectre of Häggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume G has a 2-factor with at least k+1 odd components. Then G is hamiltonian
Long cycles, degree sums and neighborhood unions
AbstractFor a graph G, define the parameters α(G)=max{|S| |S is an independent set of vertices of G}, σk(G)=min{∑ki=1d(vi)|{v1,…,vk} is an independent set} and NCk(G)= min{|∪ki=1 N(vi)∥{v1,…,vk} is an independent set} (k⩾2). It is shown that every 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,n+NCr+5+∈(n+r)(G)-α(G)}, where ε(i)=3(⌈13i⌉−13i). This result extends previous results in Bauer et al. (1989/90), Faßbender (1992) and Flandrin et al. (1991). It is also shown that a 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,2NC⌊18(n+6r+17)⌋(G)}. Analogous results are established for 2-connected graphs
Limit Cycle Bifurcations from Centers of Symmetric Hamiltonian Systems Perturbing by Cubic Polynomials
In this paper, we consider some cubic near-Hamiltonian systems obtained from
perturbing the symmetric cubic Hamiltonian system with two symmetric singular
points by cubic polynomials. First, following Han [2012] we develop a method to
study the analytical property of the Melnikov function near the origin for
near-Hamiltonian system having the origin as its elementary center or nilpotent
center. Based on the method, a computationally efficient algorithm is
established to systematically compute the coefficients of Melnikov function.
Then, we consider the symmetric singular points and present the conditions for
one of them to be elementary center or nilpotent center. Under the condition
for the singular point to be a center, we obtain the normal form of the
Hamiltonian systems near the center. Moreover, perturbing the symmetric cubic
Hamiltonian systems by cubic polynomials, we consider limit cycles bifurcating
from the center using the algorithm to compute the coefficients of Melnikov
function. Finally, perturbing the symmetric hamiltonian system by symmetric
cubic polynomials, we consider the number of limit cycles near one of the
symmetric centers of the symmetric near-Hamiltonian system, which is same to
that of another center
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
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