The main contribution of this paper is a new approach for enumerating Hamilton cycles in bounded degree graphs – deriving thereby extremal bounds. We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular n-vertex graph in time O(1.276), improving on Eppstein’s previous bound. The resulting new upper bound of O(1.276) for the maximum number of Hamilton cycles in 3-regul...

Some bipartite Hamilton decomposable graphs that are regular of degree δ ≡ 2 (mod 4) are shown to have Hamilton decomposable line graphs. One consequence is that every bipartite Hamilton decomposable graph G with connectivity κ(G) = 2 has a Hamilton decomposable line graph L(G).

We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the th...

Multiple edge-disjoint Hamilton cycles have been obtained in labelled star graphs Stn of degree n-1, using number-theoretic means, as images of a known base 2-labelled Hamilton cycle under label-mapping automorphisms of Stn. However, no optimum bounds for producing such edge-disjoint Hamilton cycles have been given, and no positive or negative results exist on whether Hamilton decompositions ca...

A graph G is 1-Hamilton-connected if G − x is Hamilton-connected for every vertex x ∈ V (G). In the paper we introduce a closure concept for 1-Hamiltonconnectedness in claw-free graphs. If G is a (new) closure of a claw-free graph G, then G is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, G is the line graph of a multigraph, and for some x ∈ V (G), G − x is the line graph of a ...

A graph G is 1-Hamilton-connected if G − x is Hamilton-connected for every x ∈ V (G), and G is 2-edge-Hamilton-connected if the graph G + X has a hamiltonian cycle containing all edges of X for any X ⊂ E+(G) = {xy| x, y ∈ V (G)} with 1 ≤ |X| ≤ 2. We prove that Thomassen’s conjecture (every 4-connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent...

The paper is concerned with the Hamilton-Jacobi (HJ) equations of multidimensional space variables with convex initial data and general Hamiltonians. Using Hopf’s formula (II), we will study the differentiability of the HJ solutions. For any given point,we give a sufficient and necessary condition such that the solutions are Ck smooth in some neighborhood of this point. We also study the charac...

This paper is concerned with the Hamilton-Jacobi (HJ) equations of multidimensional space variables with convex Hamiltonian. Using Hopf’s formula (I), we will study the differentiability of the HJ solutions. For any given point, we give a sufficient and necessary condition under which the solutions are Ck smooth in some neighborhood of the point. We also study the characteristics of the HJ equa...