51,312 research outputs found
Many toric ideals generated by quadratic binomials possess no quadratic Gr\"obner bases
Let be a finite connected simple graph and the toric ideal of the
edge ring of . In the present paper, we study finite graphs with
the property that is generated by quadratic binomials and
possesses no quadratic Gr\"obner basis. First, we give a nontrivial infinite
series of finite graphs with the above property. Second, we implement a
combinatorial characterization for to be generated by quadratic
binomials and, by means of the computer search, we classify the finite graphs
with the above property, up to 8 vertices.Comment: 11 pages, 17 figures, Typos corrected, Reference adde
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
Graph-theoretic conditions for injectivity of functions on rectangular domains
This paper presents sufficient graph-theoretic conditions for injectivity of
collections of differentiable functions on rectangular subsets of R^n. The
results have implications for the possibility of multiple fixed points of maps
and flows. Well-known results on systems with signed Jacobians are shown to be
easy corollaries of more general results presented here.Comment: 16 pages, 5 figure
Some local--global phenomena in locally finite graphs
In this paper we present some results for a connected infinite graph with
finite degrees where the properties of balls of small radii guarantee the
existence of some Hamiltonian and connectivity properties of . (For a vertex
of a graph the ball of radius centered at is the subgraph of
induced by the set of vertices whose distance from does not
exceed ). In particular, we prove that if every ball of radius 2 in is
2-connected and satisfies the condition for
each path in , where and are non-adjacent vertices, then
has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017).
Furthermore, we prove that if every ball of radius 1 in satisfies Ore's
condition (1960) then all balls of any radius in are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio
On the order of countable graphs
A set of graphs is said to be independent if there is no homomorphism between
distinct graphs from the set. We consider the existence problems related to the
independent sets of countable graphs. While the maximal size of an independent
set of countable graphs is 2^omega the On Line problem of extending an
independent set to a larger independent set is much harder. We prove here that
singletons can be extended (``partnership theorem''). While this is the best
possible in general, we give structural conditions which guarantee independent
extensions of larger independent sets. This is related to universal graphs,
rigid graphs and to the density problem for countable graphs
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