68,383 research outputs found
A degree sum condition for graphs to be covered by two cycles
AbstractLet G be a k-connected graph of order n. In [1], Bondy (1980) considered a degree sum condition for a graph to have a Hamiltonian cycle, say, to be covered by one cycle. He proved that if σk+1(G)>(k+1)(n−1)/2, then G has a Hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. (1995) [4] proved that if k=1 and σ3(G)≥n, then G can be covered by two cycles. By these results, we conjecture that if σ2k+1(G)>(2k+1)(n−1)/3, then G can be covered by two cycles. In this paper, we prove the case k=2 of this conjecture. In fact, we prove a stronger result; if G is 2-connected with σ5(G)≥5(n−1)/3, then G can be covered by two cycles, or G belongs to an exceptional class
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
Long paths and cycles in random subgraphs of graphs with large minimum degree
For a given finite graph of minimum degree at least , let be a
random subgraph of obtained by taking each edge independently with
probability . We prove that (i) if for a function
that tends to infinity as does, then
asymptotically almost surely contains a cycle (and thus a path) of length at
least , and (ii) if , then
asymptotically almost surely contains a path of length at least . Our
theorems extend classical results on paths and cycles in the binomial random
graph, obtained by taking to be the complete graph on vertices.Comment: 26 page
Toric algebra of hypergraphs
The edges of any hypergraph parametrize a monomial algebra called the edge
subring of the hypergraph. We study presentation ideals of these edge subrings,
and describe their generators in terms of balanced walks on hypergraphs. Our
results generalize those for the defining ideals of edge subrings of graphs,
which are well-known in the commutative algebra community, and popular in the
algebraic statistics community. One of the motivations for studying toric
ideals of hypergraphs comes from algebraic statistics, where generators of the
toric ideal give a basis for random walks on fibers of the statistical model
specified by the hypergraph. Further, understanding the structure of the
generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in
algebraic statistics and to combinatorial discrepancy. Section 6 (open
problems) has been moderately revise
A simple model of trees for unicellular maps
We consider unicellular maps, or polygon gluings, of fixed genus. A few years
ago the first author gave a recursive bijection transforming unicellular maps
into trees, explaining the presence of Catalan numbers in counting formulas for
these objects. In this paper, we give another bijection that explicitly
describes the "recursive part" of the first bijection. As a result we obtain a
very simple description of unicellular maps as pairs made by a plane tree and a
permutation-like structure. All the previously known formulas follow as an
immediate corollary or easy exercise, thus giving a bijective proof for each of
them, in a unified way. For some of these formulas, this is the first bijective
proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and
the Goupil-Schaeffer formula. We also discuss several applications of our
construction: we obtain a new proof of an identity related to covered maps due
to Bernardi and the first author, and thanks to previous work of the second
author, we give a new expression for Stanley character polynomials, which
evaluate irreducible characters of the symmetric group. Finally, we show that
our techniques apply partially to unicellular 3-constellations and to related
objects that we call quasi-constellations.Comment: v5: minor revision after reviewers comments, 33 pages, added a
refinement by degree of the Harer-Zagier formula and more details in some
proof
Edge-disjoint Hamilton cycles in graphs
In this paper we give an approximate answer to a question of Nash-Williams
from 1970: we show that for every \alpha > 0, every sufficiently large graph on
n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8
edge-disjoint Hamilton cycles. More generally, we give an asymptotically best
possible answer for the number of edge-disjoint Hamilton cycles that a graph G
with minimum degree \delta must have. We also prove an approximate version of
another long-standing conjecture of Nash-Williams: we show that for every
\alpha > 0, every (almost) regular and sufficiently large graph on n vertices
with minimum degree at least can be almost decomposed into
edge-disjoint Hamilton cycles.Comment: Minor Revisio
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