137,170 research outputs found
Counting Independent Sets of a Fixed Size in Graphs with Given Minimum Degree
Galvin showed that for all fixed δ and sufficiently large n, the n-vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph . He conjectured that except perhaps for some small values of t, the same graph yields the maximum count of independent sets of size t for each possible t. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples with , no n-vertex bipartite graph with minimum degree δ admits more independent sets of size t than . Here, we make further progress. We show that for all triples with and , no n-vertex graph with minimum degree δ admits more independent sets of size t than , and we obtain the same conclusion for and . Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Dense H-free graphs are almost (Χ(H)-1)-partite
By using the Szemeredi Regularity Lemma, Alon and Sudakov recently
extended the classical Andrasfai-Erdos-Sos theorem to cover general graphs. We
prove, without using the Regularity Lemma, that the following stronger statement
is true.
Given any (r+1)-partite graph H whose smallest part has t vertices, there exists
a constant C such that for any given ε>0 and sufficiently large n the following is
true. Whenever G is an n-vertex graph with minimum degree
δ(G)≥(1 −
3/3r−1 + ε)n,
either G contains H, or we can delete f(n,H)≤Cn2−1/t edges from G to obtain an
r-partite graph. Further, we are able to determine the correct order of magnitude
of f(n,H) in terms of the Zarankiewicz extremal function
Chromatic thresholds in dense random graphs
The chromatic threshold of a graph with respect to the
random graph is the infimum over such that the following holds
with high probability: the family of -free graphs with
minimum degree has bounded chromatic number. The study of
the parameter was initiated in 1973 by
Erd\H{o}s and Simonovits, and was recently determined for all graphs . In
this paper we show that for all fixed , but that typically if . We also make significant progress towards determining
for all graphs in the range . In sparser random graphs the
problem is somewhat more complicated, and is studied in a separate paper.Comment: 36 pages (including appendix), 1 figure; the appendix is copied with
minor modifications from arXiv:1108.1746 for a self-contained proof of a
technical lemma; accepted to Random Structures and Algorithm
Maximizing the number of independent sets of a fixed size
Let be the number of independent sets of size in a graph .
Engbers and Galvin asked how large could be in graphs with minimum
degree at least . They further conjectured that when
and , is maximized by the complete bipartite graph
. This conjecture has drawn the attention of many
researchers recently. In this short note, we prove this conjecture.Comment: 5 page
- …