3,942 research outputs found
Heavy subgraphs, stability and hamiltonicity
Let be a graph. Adopting the terminology of Broersma et al. and \v{C}ada,
respectively, we say that is 2-heavy if every induced claw () of
contains two end-vertices each one has degree at least ; and
is o-heavy if every induced claw of contains two end-vertices with degree
sum at least in . In this paper, we introduce a new concept, and
say that is \emph{-c-heavy} if for a given graph and every induced
subgraph of isomorphic to and every maximal clique of ,
every non-trivial component of contains a vertex of degree at least
in . In terms of this concept, our original motivation that a
theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and
-c-heavy graph is hamiltonian, where is the graph obtained from a
triangle by adding three disjoint pendant edges. In this paper, we will
characterize all connected graphs such that every 2-connected o-heavy and
-c-heavy graph is hamiltonian. Our work results in a different proof of a
stronger version of Hu's theorem. Furthermore, our main result improves or
extends several previous results.Comment: 21 pages, 6 figures, finial version for publication in Discussiones
Mathematicae Graph Theor
Construction of Maximal Hypersurfaces with Boundary Conditions
We construct maximal hypersurfaces with a Neumann boundary condition in
Minkowski space via mean curvature flow. In doing this we give general
conditions for long time existence of the flow with boundary conditions with
assumptions on the curvature of a the Lorentz boundary manifold
The typical structure of maximal triangle-free graphs
Recently, settling a question of Erd\H{o}s, Balogh and
Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most
-vertex maximal triangle-free graphs, matching the previously known lower
bound. Here we characterize the typical structure of maximal triangle-free
graphs. We show that almost every maximal triangle-free graph admits a
vertex partition such that is a perfect matching and is an
independent set.
Our proof uses the Ruzsa-Szemer\'{e}di removal lemma, the
Erd\H{o}s-Simonovits stability theorem, and recent results of
Balogh-Morris-Samotij and Saxton-Thomason on characterization of the structure
of independent sets in hypergraphs. The proof also relies on a new bound on the
number of maximal independent sets in triangle-free graphs with many
vertex-disjoint 's, which is of independent interest.Comment: 17 page
Upper bound theorem for odd-dimensional flag triangulations of manifolds
We prove that among all flag triangulations of manifolds of odd dimension
2r-1 with sufficiently many vertices the unique maximizer of the entries of the
f-, h-, g- and gamma-vector is the balanced join of r cycles. Our proof uses
methods from extremal graph theory.Comment: Clarifications and new references, title has change
Graph properties, graph limits and entropy
We study the relation between the growth rate of a graph property and the
entropy of the graph limits that arise from graphs with that property. In
particular, for hereditary classes we obtain a new description of the colouring
number, which by well-known results describes the rate of growth.
We study also random graphs and their entropies. We show, for example, that
if a hereditary property has a unique limiting graphon with maximal entropy,
then a random graph with this property, selected uniformly at random from all
such graphs with a given order, converges to this maximizing graphon as the
order tends to infinity.Comment: 24 page
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