649 research outputs found
Two problems on independent sets in graphs
Let denote the number of independent sets of size in a graph
. Levit and Mandrescu have conjectured that for all bipartite the
sequence (the {\em independent set sequence} of ) is
unimodal. We provide evidence for this conjecture by showing that is true for
almost all equibipartite graphs. Specifically, we consider the random
equibipartite graph , and show that for any fixed its
independent set sequence is almost surely unimodal, and moreover almost surely
log-concave except perhaps for a vanishingly small initial segment of the
sequence. We obtain similar results for .
We also consider the problem of estimating for
in various families. We give a sharp upper bound on the number of
independent sets in an -vertex graph with minimum degree , for all
fixed and sufficiently large . Specifically, we show that the
maximum is achieved uniquely by , the complete bipartite
graph with vertices in one partition class and in the
other.
We also present a weighted generalization: for all fixed and , as long as is large enough, if is a graph on
vertices with minimum degree then with equality if and only if
.Comment: 15 pages. Appeared in Discrete Mathematics in 201
Extremal \u3cem\u3eH\u3c/em\u3e-Colorings of Trees and 2-connected Graphs
For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several results in this area. First, we find a class of graphs H with the property that for each H∈H, the n-vertex tree that minimizes the number of H -colorings is the path Pn. We then present a new proof of a theorem of Sidorenko, valid for large n, that for every H the star K1,n−1 is the n-vertex tree that maximizes the number of H-colorings. Our proof uses a stability technique which we also use to show that for any non-regular H (and certain regular H ) the complete bipartite graph K2,n−2 maximizes the number of H-colorings of n -vertex 2-connected graphs. Finally, we show that the cycle Cn has the most proper q-colorings among all n-vertex 2-connected graphs
- …