On stability of the Hamiltonian index under contractions and closures

The hamiltonian index of a graph $G$ is the smallest integer $k$ such that the $k$-th iterated line graph of $G$ is hamiltonian. We first show that, with one exceptional case, adding an edge to a graph cannot increase its hamiltonian index. We use this result to prove that neither the contraction of an $A_G(F)$-contractible subgraph $F$ of a graph $G$ nor the closure operation performed on $G$ (if $G$ is claw-free) affects the value of the hamiltonian index of a graph $G$

The Hamiltonian index of a graph and its branch-bonds

Let $G$ be an undirected and loopless finite graph that is not a path. The minimum $m$ such that the iterated line graph $L^m(G)$ is hamiltonian is called the hamiltonian index of $G,$ denoted by $h(G).$ A reduction method to determine the hamiltonian index of a graph $G$ with $h(G)\geq 2$ is given here. With it we will establish a sharp lower bound and a sharp upper bound for $h(G)$, respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that $h(G)$ may reach all integers between the lower bound and the upper bound. \u

Computing and counting longest paths on circular-arc graphs in polynomial time.

The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G′ of G admits a path P′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight

Uniqueness of canonical tensor model with local time

Canonical formalism of the rank-three tensor model has recently been proposed, in which "local" time is consistently incorporated by a set of first class constraints. By brute-force analysis, this paper shows that there exist only two forms of a Hamiltonian constraint which satisfies the following assumptions: (i) A Hamiltonian constraint has one index. (ii) The kinematical symmetry is given by an orthogonal group. (iii) A consistent first class constraint algebra is formed by a Hamiltonian constraint and the generators of the kinematical symmetry. (iv) A Hamiltonian constraint is invariant under time reversal transformation. (v) A Hamiltonian constraint is an at most cubic polynomial function of canonical variables. (vi) There are no disconnected terms in a constraint algebra. The two forms are the same except for a slight difference in index contractions. The Hamiltonian constraint which was obtained in the previous paper and behaved oddly under time reversal symmetry can actually be transformed to one of them by a canonical change of variables. The two-fold uniqueness is shown up to the potential ambiguity of adding terms which vanish in the limit of pure gravitational physics.Comment: 21 pages, 12 figures. The final result unchanged. Section 5 rewritten for clearer discussions. The range of uniqueness commented in the final section. Some other minor correction

12, 24 and Beyond

We generalize the well-known "12" and "24" Theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a wider category of objects, here called reflexive GKM graphs, that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action. As an application, we provide bounds on the Betti numbers for certain monotone Hamiltonian spaces which depend on the minimal Chern number of the manifold.Comment: 39 pages, 4 figure