We establish the relation between the ISO(2, 1) homotopy invariants, and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2, 1) cycle conditions, are equivalent to the ISO(2, 1) cycle conditions for the representations ρ : π 1 (Σ g,N) → ISO(2, 1). Also, the symplectic structure on the space of invariants is closely related to that of ...

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p (log k + log log k + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph ...

For two integers n ≥ 3 and 2 ≤ p ≤ n, we denote D(n, p) the digraph obtained from a directed n-cycle by changing the orientations of p − 1 consecutive arcs. In this paper, we show that a family of k-regular (k ≥ 3) bipartite tournament BT4k contains D(4k, p) for all 2 ≤ p ≤ 4k unless BT4k is isomorphic to a digraph D such that (1, 2, 3, ..., 4k, 1) is a Hamilton cycle and (4m+ i− 1, i) ∈ A(D) a...

We study Hamiltonicity in graphs obtained as the union of a deterministic $n$-vertex graph $H$ with linear degrees and $d$-dimensional random geometric $G^d(n,r)$, for any $d\geq1$. obtain an asymptotically optimal bound on minimum $r$ which a.a.s. $H\cup G^d(n,r)$ is Hamiltonian. Our proof provides time algorithm to find Hamilton cycle such graphs.

We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. Recall every Hamiltonian graph is connected, has an almost perfect matching and, excluding bipartite case, contains odd cycle. Our main result states any large enough robustly satisfies these properties must already be Hamiltonian. Moreover, same holds for powers...

New sufficient conditions are derived for generalized cycles (including Hamilton and dominating as special cases) in an arbitrary k-connected (k = 1, 2, ...) graph, which prove the truth of Bondy’s (1980) famous conjecture some variants significantly improving result expected by given hypothesis. Similarly, new lower bounds circumference (the length a longest cycle) established reverse hypothes...

We construct new resolvable decompositions of complete multigraphs and equipartite into cycles variable lengths (and a perfect matching if the vertex degrees are odd). develop two techniques: layering, which allows us to obtain 2-factorizations from existing graphs, detachment, cycle multigraphs. These techniques applied specified type for both multigraphs, with emphasis on solutions Oberwolfac...

1 Hamiltonian Cycle “Non-local” problems can also be NP-hard. Recall that a Hamilton cycle is one where all vertices is visited exactly once. Name: HC Input: A graph G. Output: Does G contain a Hamiltonian cycle? In other words, HC is asking whether the longest cycle in G has length n. This problem is “non-local” in the sense that the constraint involves all vertices (a.k.a. global). Theorem 1....

The main results of this thesis are the following. We show that for each α > 0 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least 3|G|/8 + α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. Furthermore, answering completely a conjecture of Häggkvist and Thomason, we show that we get every possible orientation...

A graph G is said to be claw-free if G has no induced subgraph isomorphic to K1,3. For a cycle C in a graph G, C is called a Tutte cycle of G if C is a Hamilton cycle of G, or the order of C is at least 4 and every component of G − C has at most three neighbors on C. In [On a closure concept in claw-free graphs, J. Combin. Theory Ser. B 70 (1997), 217–224], Ryjáček proved that the conjectures b...