Enumeration of Hamiltonian cycles in certain generalized Petersen graphs

AbstractThe generalized Petersen graph P(n, k) has vertex set V={u0, u1, …, un−1, v0, v1, …, vn−1} and edge set E={uiui+1, uivi, vivi+k∥ for 0≤i≤n−1 with indices taken modulo n}. The classification of the Hamiltonicity of generalized Petersen graphs was begun by Watkins, continued by Bondy and Bannai, and completed by Alspach. We now determine the precise number of Hamiltonian cycles present in each of the graphs P(n, 2). This more detailed information allows us to identify an infinite family of counterexamples to a conjecture of Greenwell and Kronk who had suggested a relation between uniquely 3-edge-colorable cubic graphs and the number of Hamiltonian cycles present

Triangle-free Uniquely 3-Edge Colorable Cubic Graphs

This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976

Recognizing Graph Theoretic Properties with Polynomial Ideals

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Groebner bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure

Triangle-free Uniquely 3-Edge Colorable Cubic Graphs

This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs.  The only such graph previously known was given by Tutte in 1976

Spanning cycles of nearly cubic graphs

AbstractThe fact that a cubic hamiltonian graph must have at least three spanning cycles suggests the question of whether every hamiltonian graph in which each point has degree at least 3 must have at least three spanning cycles. We answer this in the negative by exhibiting graphs on n=2m+1, m≥5, points in which one point has degree 4, all others have degree 3, and only two spanning cycles exist

A Victorian Age Proof of the Four Color Theorem

In this paper we have investigated some old issues concerning four color map problem. We have given a general method for constructing counter-examples to Kempe's proof of the four color theorem and then show that all counterexamples can be rule out by re-constructing special 2-colored two paths decomposition in the form of a double-spiral chain of the maximal planar graph. In the second part of the paper we have given an algorithmic proof of the four color theorem which is based only on the coloring faces (regions) of a cubic planar maps. Our algorithmic proof has been given in three steps. The first two steps are the maximal mono-chromatic and then maximal dichromatic coloring of the faces in such a way that the resulting uncolored (white) regions of the incomplete two-colored map induce no odd-cycles so that in the (final) third step four coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio