28,533 research outputs found
Generalized Gr\"otzsch Graphs
The aim of this paper is to present a generalization of Gr\"otzsch graph.
Inspired by structure of the Gr\"otzsch's graph, we present constructions of
two families of graphs, and for odd and even values of
respectively and on vertices. We show that each member of this
family is non-planar, triangle-free, and Hamiltonian. Further, when is odd
the graph is maximal triangle-free, and when is even, the addition of
exactly edges makes the graph maximal triangle-free. We
show that is 4-chromatic and is 3-chromatic for all . Further,
we note some other properties of these graphs and compare with Mycielski's
construction.Comment: This is a first draft report about ongoing work on the Gr\"otzsch
Graph
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
Integer symmetric matrices having all their eigenvalues in the interval [-2,2]
We completely describe all integer symmetric matrices that have all their
eigenvalues in the interval [-2,2]. Along the way we classify all signed
graphs, and then all charged signed graphs, having all their eigenvalues in
this same interval. We then classify subsets of the above for which the integer
symmetric matrices, signed graphs and charged signed graphs have all their
eigenvalues in the open interval (-2,2).Comment: 33 pages, 18 figure
- …