286,546 research outputs found
The Number of Independent Sets in a Regular Graph
We show that the number of independent sets in an N-vertex, d-regular graph
is at most (2^{d+1} - 1)^{N/2d}, where the bound is sharp for a disjoint union
of complete d-regular bipartite graphs. This settles a conjecture of Alon in
1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be
bipartite. We give a short proof that reduces the general case to the bipartite
case. Our method also works for a weighted generalization, i.e., an upper bound
for the independence polynomial of a regular graph.Comment: 5 pages. Accepted by Combin. Probab. Compu
Counting Independent Sets and Colorings on Random Regular Bipartite Graphs
We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. In the weighted case, for all sufficiently large integers Delta and weight parameters lambda = Omega~ (1/(Delta)), we also obtain an FPTAS on almost every Delta-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all q >= 3 and sufficiently large integers Delta=Delta(q), there is an FPTAS to count the number of q-colorings on almost every Delta-regular bipartite graph
The Bipartite Swapping Trick on Graph Homomorphisms
We provide an upper bound to the number of graph homomorphisms from to
, where is a fixed graph with certain properties, and varies over
all -vertex, -regular graphs. This result generalizes a recently resolved
conjecture of Alon and Kahn on the number of independent sets. We build on the
work of Galvin and Tetali, who studied the number of graph homomorphisms from
to when is bipartite. We also apply our techniques to graph
colorings and stable set polytopes.Comment: 22 pages. To appear in SIAM J. Discrete Mat
Minimizing the number of independent sets in triangle-free regular graphs
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the
result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the
independence polynomial of a -regular graph is maximized by disjoint copies
of . Their proof uses linear programming bounds on the distribution of
a cleverly chosen random variable. In this paper, we use this method to give
lower bounds on the independence polynomial of regular graphs. We also give new
bounds on the number of independent sets in triangle-free regular graphs
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