32 research outputs found
Note on the smallest root of the independence polynomial
One can define the independence polynomial of a graph G as follows. Let i(k)(G) denote the number of independent sets of size k of G, where i(0)(G) = 1. Then the independence polynomial of G is I(G,x) = Sigma(n)(k=0)(-1)(k)i(k)(G)x(k). In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real
Maximal antichains of minimum size
Let be a natural number, and let be a set . We study the problem to find the smallest possible size of a
maximal family of subsets of such that
contains only sets whose size is in , and for all
, i.e. is an antichain. We present a
general construction of such antichains for sets containing 2, but not 1.
If our construction asymptotically yields the smallest possible size
of such a family, up to an error. We conjecture our construction to be
asymptotically optimal also for , and we prove a weaker bound for
the case . Our asymptotic results are straightforward applications of
the graph removal lemma to an equivalent reformulation of the problem in
extremal graph theory which is interesting in its own right.Comment: fixed faulty argument in Section 2, added reference
On the Fon-der-Flaass Interpretation of Extremal Examples for Turan's (3,4)-problem
In 1941, Turan conjectured that the edge density of any 3-graph without
independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he
gave the first example witnessing this bound. Brown (1983) and Kostochka (1982)
found many other examples of this density. Fon-der-Flaass (1988) presented a
general construction that converts an arbitrary -free orgraph
into a Turan (3,4)-graph. He observed that all Turan-Brown-Kostochka
examples result from his construction, and proved the bound >= 3/7(1-o(1)) on
the edge density of any Turan (3,4)-graph obtainable in this way.
In this paper we establish the optimal bound 4/9(1-o(1)) on the edge density
of any Turan (3,4)-graph resulting from the Fon-der-Flaass construction under
any of the following assumptions on the undirected graph underlying the
orgraph :
1. is complete multipartite;
2. The edge density of is >= (2/3-epsilon) for some absolute constant
epsilon>0.
We are also able to improve Fon-der-Flaass's bound to 7/16(1-o(1)) without
any extra assumptions on
Cycles of length three and four in tournaments
Linial and Morgenstern conjectured that, among all -vertex tournaments
with cycles of length three, the number of cycles of length
four is asymptotically minimized by a random blow-up of a transitive tournament
with all but one part of equal size and one smaller part. We prove the
conjecture for by analyzing the possible spectrum of adjacency
matrices of tournaments. We also demonstrate that the family of extremal
examples is broader than expected and give its full description for
On the number of edge-disjoint triangles in K4-free graphs
We prove the quarter of a century old conjecture of ErdĆs that every K4-free graph with n vertices and ân2/4â+m edges contains m pairwise edge disjoint triangles. © 2017 Elsevier B.V
Asymptotic Structure of Graphs with the Minimum Number of Triangles
We consider the problem of minimizing the number of triangles in a graph of
given order and size and describe the asymptotic structure of extremal graphs.
This is achieved by characterizing the set of flag algebra homomorphisms that
minimize the triangle density.Comment: 22 pages; 2 figure