13 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
The Lefthanded Local Lemma characterizes chordal dependency graphs
Shearer gave a general theorem characterizing the family \LLL of dependency
graphs labeled with probabilities which have the property that for any
family of events with a dependency graph from \LLL (whose vertex-labels are
upper bounds on the probabilities of the events), there is a positive
probability that none of the events from the family occur.
We show that, unlike the standard Lov\'asz Local Lemma---which is less
powerful than Shearer's condition on every nonempty graph---a recently proved
`Lefthanded' version of the Local Lemma is equivalent to Shearer's condition
for all chordal graphs. This also leads to a simple and efficient algorithm to
check whether a given labeled chordal graph is in \LLL.Comment: 12 pages, 1 figur
Finitely dependent coloring
We prove that proper coloring distinguishes between block-factors and
finitely dependent stationary processes. A stochastic process is finitely
dependent if variables at sufficiently well-separated locations are
independent; it is a block-factor if it can be expressed as an equivariant
finite-range function of independent variables. The problem of finding
non-block-factor finitely dependent processes dates back to 1965. The first
published example appeared in 1993, and we provide arguably the first natural
examples. More precisely, Schramm proved in 2008 that no stationary 1-dependent
3-coloring of the integers exists, and conjectured that no stationary
k-dependent q-coloring exists for any k and q. We disprove this by constructing
a 1-dependent 4-coloring and a 2-dependent 3-coloring, thus resolving the
question for all k and q.
Our construction is canonical and natural, yet very different from all
previous schemes. In its pure form it yields precisely the two finitely
dependent colorings mentioned above, and no others. The processes provide
unexpected connections between extremal cases of the Lovasz local lemma and
descent and peak sets of random permutations. Neither coloring can be expressed
as a block-factor, nor as a function of a finite-state Markov chain; indeed, no
stationary finitely dependent coloring can be so expressed. We deduce
extensions involving d dimensions and shifts of finite type; in fact, any
non-degenerate shift of finite type also distinguishes between block-factors
and finitely dependent processes
Sequences with changing dependencies
Consider words over an alphabet with n letters. Fisher [Amer. Math. Monthly, 96 (1989), pp. 610 - 614] calculated the number of distinct words of length l assuming certain pairs of letters commute. In this paper we are interested in a more general setting where the pairs of letters that commute at a certain position of a word depend on the initial segment of the word. In particular, we show that if for each word at each position any letter fails to commute with at most a constant number of other letters, then the number of distinct words of length l is at most Cn+l for some constant C. We use this result to obtain a lower bound on the number of diagonal flips required in the worst case to transform one n-vertex labeled triangulated planar graph into some other one
Bears with Hats and Independence Polynomials
Consider the following hat guessing game. A bear sits on each vertex of a
graph , and a demon puts on each bear a hat colored by one of colors.
Each bear sees only the hat colors of his neighbors. Based on this information
only, each bear has to guess colors and he guesses correctly if his hat
color is included in his guesses. The bears win if at least one bear guesses
correctly for any hat arrangement.
We introduce a new parameter - fractional hat chromatic number ,
arising from the hat guessing game. The parameter is related to the
hat chromatic number which has been studied before. We present a surprising
connection between the hat guessing game and the independence polynomial of
graphs. This connection allows us to compute the fractional hat chromatic
number of chordal graphs in polynomial time, to bound fractional hat chromatic
number by a function of maximum degree of , and to compute the exact value
of of cliques, paths, and cycles