45 research outputs found
Ear-Slicing for Matchings in Hypergraphs
We study when a given edge of a factor-critical graph is contained in a
matching avoiding exactly one, pregiven vertex of the graph. We then apply the
results to always partition the vertex-set of a -regular, -uniform
hypergraph into at most one triangle (hyperedge of size ) and edges (subsets
of size of hyperedges), corresponding to the intuition, and providing new
insight to triangle and edge packings of Cornu\'ejols' and Pulleyblank's. The
existence of such a packing can be considered to be a hypergraph variant of
Petersen's theorem on perfect matchings, and leads to a simple proof for a
sharpening of Lu's theorem on antifactors of graphs
Computation of Contour Integrals on
Contour integrals of rational functions over , the moduli
space of -punctured spheres, have recently appeared at the core of the
tree-level S-matrix of massless particles in arbitrary dimensions. The contour
is determined by the critical points of a certain Morse function on . The integrand is a general rational function of the puncture
locations with poles of arbitrary order as two punctures coincide. In this note
we provide an algorithm for the analytic computation of any such integral. The
algorithm uses three ingredients: an operation we call general KLT, Petersen's
theorem applied to the existence of a 2-factor in any 4-regular graph and
Hamiltonian decompositions of certain 4-regular graphs. The procedure is
iterative and reduces the computation of a general integral to that of simple
building blocks. These are integrals which compute double-color-ordered partial
amplitudes in a bi-adjoint cubic scalar theory.Comment: 36+11 p
The matching polynomial of a regular graph
AbstractThe matching polynomial of a graph has coefficients that give the number of matchings in the graph. For a regular graph, we show it is possible to recover the order, degree, girth and number of minimal cycles from the matching polynomial. If a graph is characterized by its matching polynomial, then it is called matching unique. Here we establish the matching uniqueness of many specific regular graphs; each of these graphs is either a cage, or a graph whose components are isomorphic to Moore graphs. Our main tool in establishing the matching uniqueness of these graphs is the ability to count certain subgraphs of a regular graph
Every graph occurs as an induced subgraph of some hypohamiltonian graph
We prove the titular statement. This settles a problem of Chvátal from 1973 and encompasses earlier results of Thomassen, who showed it for K_3, and Collier and Schmeichel, who proved it for bipartite graphs. We also show that for every outerplanar graph there exists a planar hypohamiltonian graph containing it as an induced subgraph
Even cycles with prescribed chords in planar cubic graphs
AbstractThe following result is being proved. Theorem: Let e be an arbitrary line of the 2-connected, 3-regular, planar graph G such that e does not belong to any cut set of size 2. The G contains an even cycle for which e is a chord