14,881 research outputs found
A graph polynomial for independent sets of bipartite graphs
We introduce a new graph polynomial that encodes interesting properties of
graphs, for example, the number of matchings and the number of perfect
matchings. Most importantly, for bipartite graphs the polynomial encodes the
number of independent sets (#BIS).
We analyze the complexity of exact evaluation of the polynomial at rational
points and show that for most points exact evaluation is #P-hard (assuming the
generalized Riemann hypothesis) and for the rest of the points exact evaluation
is trivial.
We conjecture that a natural Markov chain can be used to approximately
evaluate the polynomial for a range of parameters. The conjecture, if true,
would imply an approximate counting algorithm for #BIS, a problem shown, by
[Dyer et al. 2004], to be complete (with respect to, so called, AP-reductions)
for a rich logically defined sub-class of #P. We give a mild support for our
conjecture by proving that the Markov chain is rapidly mixing on trees. As a
by-product we show that the "single bond flip" Markov chain for the random
cluster model is rapidly mixing on constant tree-width graphs
Counting independent sets in cubic graphs of given girth
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.Postprint (author's final draft
Counting independent sets in cubic graphs of given girth
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.Postprint (author's final draft
Recognizing graphs close to bipartite graphs.
We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree
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
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