Let $T$ be a non-trivial tournament. An arc is emph{$t$-pancyclic} in $T$, if it is contained in a cycle of length $ell$ for every $tleq ell leq |V(T)|$. Let $p^t(T)$ denote the number of $t$-pancyclic arcs in $T$ and $h^t(T)$ the maximum number of $t$-pancyclic arcs contained in the same Hamiltonian cycle of $T$. Moon ({em J. Combin. Inform. System Sci.}, {bf 19} (1994), 207-214) showed that $...

We study the Hamiltonian Cycle problem in graphs induced by subsets of the vertices of the tiling of the plane with equilateral triangles. By analogy with grid graphs we call such graphs triangular grid graphs. Following the analogy, we define the class of solid triangular grid graphs. We prove that the Hamiltonian Cycle problem is NPcomplete for triangular grid graphs. We show that with the ex...

The problem of deciding if a Traveling Salesman Problem (TSP) tour is minimal was proved to be coNP–complete by Papadimitriou and Steiglitz. We give an alternative proof based on a polynomial time reduction from 3SAT. Like the original proof, our reduction also shows that given a graph G and an Hamiltonian path of G, it is NP–complete to check if G contains an Hamiltonian cycle (Restricted Hami...

Fleischner's theorem says that the square of every 2-connected graph contains a Hamiltonian cycle. We present a proof resulting in an O(|E|) algorithm for producing a Hamiltonian cycle in the square G of a 2connected graph G = (V,E). More generally, we get an O(|E|) algorithm for producing a Hamiltonian path between any two prescribed vertices, and we get an O(|V |) algorithm for producing cycl...

We consider the spectral and algorithmic aspects of the problem of finding a Hamiltonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm is given for the problem of finding a Hamiltonian cycle in graphs with bounded spectral gaps whic...

The Hamiltonian cycle problem (HCP) in digraphs D with degree bound two is solved by two mappings in this paper. The first bijection is between an incidence matrix Cnm of simple digraph and an incidence matrix F of balanced bipartite undirected graph G; The second mapping is from a perfect matching of G to a cycle of D. It proves that the complexity of HCP in D is polynomial, and finding a seco...

A collection L = P 1 ∪ P 2 ∪ · · · ∪ P t (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V (L)| = k is called a (k, t, s)-linear forest. A graph G is (k, t, s)ordered if for every (k, t, s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k, t, s)-o...

Grinberg’s theorem is a necessary condition for the planar Hamilton graphs. In this paper, we use cycle bases and removable cycles to survey cycle structures of the Hamiltonian graphs and derive an equation of the interior faces in Grinberg’s Theorem. The result shows that Grinberg’s Theorem is suitable for the connected and simple graphs. Furthermore, by adding a new constraint of solutions to...

For MSO2-expressible problems like Edge Dominating Set or Hamiltonian Cycle, it was open for a long time whether there is an algorithm which given a clique-width k-expression of an n-vertex graph runs in time f(k) · nO(1) for some function f . Recently, Fomin et al. (SIAM. J. Computing, 2014) presented several lower bounds; for instance, there are no f(k) · n-time algorithms for Edge Dominating...

A graph is uniquely hamiltonian if it contains exactly one hamiltonian cycle. In this note we prove that there are no r-regular uniquely hamiltonian graphs when r > 22. This improves upon earlier results of Thomassen.