3,875 research outputs found
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
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
Edge-dominating cycles, k-walks and Hamilton prisms in -free graphs
We show that an edge-dominating cycle in a -free graph can be found in
polynomial time; this implies that every 1/(k-1)-tough -free graph admits
a k-walk, and it can be found in polynomial time. For this class of graphs,
this proves a long-standing conjecture due to Jackson and Wormald (1990).
Furthermore, we prove that for any \epsilon>0 every (1+\epsilon)-tough
-free graph is prism-Hamiltonian and give an effective construction of a
Hamiltonian cycle in the corresponding prism, along with few other similar
results.Comment: LaTeX, 8 page
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
Polynomial algorithms that prove an NP-hard hypothesis implies an NP-hard conclusion
A number of results in Hamiltonian graph theory are of the form implies , where is a property of graphs that is NP-hard and is a cycle structure property of graphs that is also NP-hard. Such a theorem is the well-known Chv\'{a}tal-Erd\"{o}s Theorem, which states that every graph with is Hamiltonian. Here is the vertex connectivity of and is the cardinality of a largest set of independent vertices of . In another paper Chv\'{a}tal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than independent vertices. In this note we point out that other theorems in Hamiltonian graph theory have a similar character. In particular, we present a constructive proof of the well-known theorem of Jung for graphs on or more vertices.. \u
Long cycles in graphs with large degree sums and neighborhood unions
We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings
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