1,119 research outputs found
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
A generalization of Ore's Theorem involving neighborhood unions
AbstractLet G be a graph of order n. Settling conjectures of Chen and Jackson, we prove the following generalization of Ore's Theorem: If G is 2-connected and |N(u)∪N(v)|⩾12n for every pair of nonadjacent vertices u,v, then either G is hamiltonian, or G is the Petersen graph, or G belongs to one of three families of exceptional graphs of connectivity 2
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
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
A note on dominating cycles in 2-connected graphs
Let G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z) n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs
Subgraphs, Closures and Hamiltonicity
Closure theorems in hamiltonian graph theory are of the following type: Let G be a 2- connected graph and let u, v be two distinct nonadjacent vertices of G. If condition c(u,v) holds, then G is hamiltonian if and only if G + uv is hamiltonian. We discuss several results of this type in which u and v are vertices of a subgraph H of G on four vertices and c(u, v) is a condition on the neighborhoods of the vertices of H (in G). We also discuss corresponding sufficient conditions for hamiltonicity of G
Finding long cycles in graphs
We analyze the problem of discovering long cycles inside a graph. We propose
and test two algorithms for this task. The first one is based on recent
advances in statistical mechanics and relies on a message passing procedure.
The second follows a more standard Monte Carlo Markov Chain strategy. Special
attention is devoted to Hamiltonian cycles of (non-regular) random graphs of
minimal connectivity equal to three
Parameterized Algorithms for Modular-Width
It is known that a number of natural graph problems which are FPT
parameterized by treewidth become W-hard when parameterized by clique-width. It
is therefore desirable to find a different structural graph parameter which is
as general as possible, covers dense graphs but does not incur such a heavy
algorithmic penalty.
The main contribution of this paper is to consider a parameter called
modular-width, defined using the well-known notion of modular decompositions.
Using a combination of ILPs and dynamic programming we manage to design FPT
algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path
and Hamiltonian cycle), which are W-hard for both clique-width and its recently
introduced restriction, shrub-depth. We thus argue that modular-width occupies
a sweet spot as a graph parameter, generalizing several simpler notions on
dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with
arXiv:1304.5479 by other author
The number of Hamiltonian decompositions of regular graphs
A Hamilton cycle in a graph is a cycle passing through every vertex
of . A Hamiltonian decomposition of is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph on
an odd number of vertices has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
-regular -vertex graph with even degree for some fixed
has a Hamiltonian decomposition, provided is sufficiently
large. In this paper we address the natural question of estimating ,
the number of such decompositions of . Our main result is that
. In particular, the number of Hamiltonian
decompositions of is
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