Search ResultsShowing 1-5 of 5
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arXiv:1509.05950 (Published 2015-09-20)
On the roots of hypergraph chromatic polynomials
Categories: math.COLet $G = (V,E)$ be a finite, simple, connected graph with chromatic polynomial $P_G(q)$. Sokal \cite{sokal} proved that the roots of the chromatic polynomial of $G$ are bounded in absolute value by $KD$ where, $D$ is the maximum degree of the graph and $7< K < 8$ is a constant. In this paper we generalize this result to uniform hypergraphs. To prove our results we will use the theory of the bounded exponential type graph polynomials.
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arXiv:1502.06032 (Published 2015-02-20)
A note on the shameful conjecture
Comments: Accepted to the European Journal of CombinatoricsCategories: math.COTags: journal articleLet $P_G(q)$ denote the chromatic polynomial of a graph $G$ on $n$ vertices. The `shameful conjecture' due to Bartels and Welsh states that, $$\frac{P_G(n)}{P_G(n-1)} \geq \frac{n^n}{(n-1)^n}.$$ Let $\mu(G)$ denote the expected number of colors used in a uniformly random proper $n$-coloring of $G$. The above inequality can be interpreted as saying that $\mu(G) \geq \mu(O_n)$, where $O_n$ is the empty graph on $n$ nodes. This conjecture was proved by F. M. Dong, who in fact showed that, $$\frac{P_G(q)}{P_G(q-1)} \geq \frac{q^n}{(q-1)^n}$$ for all $q \geq n$. There are examples showing that this inequality is not true for all $q \geq 2$. In this paper, we show that the above inequality holds for all $q \geq 36D^{3/2}$, where $D$ is the largest degree of $G$. It is also shown that the above inequality holds true for all $q \geq 2$ when $G$ is a claw-free graph.
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A generalization of the Birthday problem and the chromatic polynomial
Categories: math.COThe birthday paradox states that there is at least a 50% chance that some two out of twenty-three randomly chosen people will share the same birth date. The calculation for this problem assumes that all birth dates are equally likely. We consider the following two modifications of this question. If the distribution of birthdays is non-uniform, does that increase or decrease the probability of matching birth dates? Further, what if we focus on birthdays shared by some particular pairs rather than any two people. Does a non-uniform distribution on birth dates increase or decrease the probability of a matching pair? In this paper we present our results in this generalized setting. We use some results and methods due to Sokal concerning bounds on the roots of chromatic polynomials to prove our results.
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On Brenti's conjecture about the log-concavity of the chromatic polynomial
Categories: math.COThe chromatic polynomial is a well studied object in graph theory. There are many results and conjectures about the log-concavity of the chromatic polynomial and other polynomials related to it. The location of the roots of these polynomials has also been well studied. One famous result due to A. Sokal and C. Borgs provides a bound on the absolute value of the roots of the chromatic polynomial in terms of the highest degree of the graph. We use this result to prove a modification of a log-concavity conjecture due to F. Brenti. The original conjecture of Brenti was that the chromatic polynomial is log-concave on the natural numbers. This was disproved by Paul Seymour by presenting a counter example. We show that the chromatic polynomial $P_G(q)$ of graph $G$ is in fact log-concave for all $q > C\Delta + 1$ for an explicit constant $C < 10$, where $\Delta$ denotes the highest degree of $G$. We also provide an example which shows that the result is not true for constants $C$ smaller than 1.
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Warmth and mobility of random graphs
Comments: This version is a substantial rewrite from earlier versionsSubjects: 05C80A graph homomorphism from the rooted $d$-branching tree $\phi: T^d \to H$ is said to be cold if the values of $\phi$ for vertices arbitrarily far away from the root can restrict the value of $\phi$ at the root. Warmth is a graph parameter that measures the non-existence of cold maps. We study warmth of random graphs $G(n,p)$, and for every $d \ge 1$, we exhibit a nearly-sharp threshold for the existence of cold maps. As a corollary, for $p=O(n^{-\alpha})$ warmth of $G(n,p)$ is concentrated on at most two values. As another corollary, a conjecture of Lov\'asz relating mobility to chromatic number holds for "almost all" graphs. Finally, our results suggest new conjectures relating graph parameters from statistical physics with graph parameters from equivariant topology.