Showing papers in "Electronic Journal of Combinatorics in 2004"

Planar Maps as Labeled Mobiles

Abstract: We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.

A Complete Annotated Bibliography of Work Related to Sidon Sequences

Abstract: A Sidon sequence is a sequence of integers $a_1 < a_2 < \cdots$ with the property that the sums $a_i + a_j$ $(i\le j)$ are distinct. This work contains a survey of Sidon sequences and their generalizations, and an extensive annotated and hyperlinked bibliography of related work.

Coloring with no $2$-Colored $P_4$'s

Abstract: A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings , i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic $k$-coloring can be refined to a star coloring with at most $(2k^2-k)$ colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are $7$-colorable, and planar graphs of girth at least $7$ are $9$-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width $t$ can be star colored with ${t+2\choose2}$ colors, and we show that this is best possible.

Short Cycles in Random Regular Graphs

Abstract: Consider random regular graphs of order $n$ and degree $d=d(n)\ge 3$ Let $g=g(n)\ge 3$ satisfy $(d-1)^{2g-1}=o(n)$ Then the number of cycles of lengths up to $g$ have a distribution similar to that of independent Poisson variables In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than $g$ A corresponding result is given for random regular bipartite graphs

Lattice Structures from Planar Graphs

Abstract: The set of all orientations of a planar graph with prescribed outdegrees carries the structure of a distributive lattice. This general theorem is proven in the first part of the paper. In the second part the theorem is applied to show that interesting combinatorial sets related to a planar graph have lattice structure: Eulerian orientations, spanning trees and Schnyder woods. For the Schnyder wood application some additional theory has to be developed. In particular it is shown that a Schnyder wood for a planar graph induces a Schnyder wood for the dual.

Pólya's Permanent Problem

Abstract: A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of sign-nonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms.

A Fast Algorithm for MacMahon's Partition Analysis

Abstract: This paper deals with evaluating constant terms of a special class of rational functions, the Elliott-rational functions. The constant term of such a function can be read off immediately from its partial fraction decomposition. We combine the theory of iterated Laurent series and a new algorithm for partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega calculus, which (partially) avoids the "run-time explosion" problem when eliminating several variables. We discuss the efficiency of our algorithm by investigating problems studied by Andrews and his coauthors; our running time is much less than that of their Omega package.

A positive proof of the Littlewood-Richardson rule using the octahedron recurrence

Abstract: We define the hive ring , which has a basis indexed by dominant weights for $GL_n({\Bbb C})$, and structure constants given by counting hives [Knutson-Tao, "The honeycomb model of $GL_n$ tensor products"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, "Involutions on Gel$'$fand-Tsetlin schemes$\dots$ "]). We use the octahedron rule from [Robbins-Rumsey, "Determinants$\dots$"] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of $GL_n({\Bbb C})$. In the honeycomb interpretation, the octahedron rule becomes "scattering" of the honeycombs. This recovers some of the "crosses and wrenches" diagrams from Speyer's very recent preprint ["Perfect matchings$\dots$"], whose results we use to give a closed form for the associativity bijection.

Vertex-Partitioning into Fixed Additive Induced-Hereditary Properties is NP-hard

Abstract: Can the vertices of an arbitrary graph $G$ be partitioned into $A \cup B$, so that $G[A]$ is a line-graph and $G[B]$ is a forest? Can $G$ be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if ${\cal P}$ and ${\cal Q}$ are additive induced-hereditary graph properties, then $({\cal P}, {\cal Q})$-colouring is NP-hard, with the sole exception of graph $2$-colouring (the case where both ${\cal P}$ and ${\cal Q}$ are the set ${\cal O}$ of finite edgeless graphs). Moreover, $({\cal P}, {\cal Q})$-colouring is NP-complete iff ${\cal P}$- and ${\cal Q}$-recognition are both in NP. This completes the proof of a conjecture of Kratochvil and Schiermeyer, various authors having already settled many sub-cases.

Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

Abstract: This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for $R_k(4)$ and $R_k(5)$ for some small $k$, including $415 \le R_3(5)$, $634 \le R_4(4)$, $2721 \le R_4(5)$, $3416 \le R_5(4)$ and $26082 \le R_5(5)$. Most of the new lower bounds are consequences of general constructions.

The Cube Recurrence

Abstract: We construct a combinatorial model that is described by the cube recurrence, a quadratic recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in ${\Bbb Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky about the structure of these polynomials, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences, including the Somos-6 and Somos-7 sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.

Sign-Graded Posets, Unimodality of $W$-Polynomials and the Charney-Davis Conjecture

Abstract: We generalize the notion of graded posets to what we call sign-graded (labeled) posets. We prove that the W-polynomial of a sign-graded poset is symmetric and unimodal. This extends a recent result of Reiner and Welker who proved it for graded posets by associating a simplicial polytopal sphere to each graded poset. By proving that the W-polynomials of sign-graded posets has the right sign at -1, we are able to prove the Charney-Davis Conjecture for these spheres (whenever they are flag).

Distinguishing Numbers for Graphs and Groups

Abstract: A graph $G$ is distinguished if its vertices are labelled by a map $\phi: V(G) \longrightarrow \{1,2,\ldots, k\}$ so that no non-trivial graph automorphism preserves $\phi$. The distinguishing number of $G$ is the minimum number $k$ necessary for $\phi$ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of $\Gamma$ on a set $X$. A labelling $\phi: X \longrightarrow \{1,2,\ldots,k\}$ is distinguishing if no element of $\Gamma$ preserves $\phi$ except those which fix each element of $X$. The distinguishing number of the group action on $X$ is the minimum $k$ needed for $\phi$ to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of $S_n$ on a set with distinguishing number $n$, answering an open question of Albertson and Collins.

Reconstructing Integer Sets From Their Representation Functions

Abstract: We give a simple common proof to recent results by Dombi and by Chen and Wang concerning the number of representations of an integer in the form $a_1+a_2$, where $a_1$ and $a_2$ are elements of a given infinite set of integers. Considering the similar problem for differences, we show that there exists a partition ${\Bbb N}=\cup_{k=1}^\infty A_k$ of the set of positive integers such that each $A_k$ is a perfect difference set (meaning that any non-zero integer has a unique representation as $a_1-a_2$ with $a_1,a_2\in A_k$). A number of open problems are presented.

Ramsey Theory Applications

Abstract: There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramsey-type theorems to various fields in mathematics are well documented in published books and monographs. The main objective of this survey is to list applications mostly in theoretical computer science of the last two decades not contained in these.

Rook theory, generalized stirling numbers and (p, q)-analogues

Abstract: In this paper, we define two natural $(p,q)$-analogues of the generalized Stirling numbers of the first and second kind $S^1(\alpha,\beta,r)$ and $S^2(\alpha,\beta,r)$ as introduced by Hsu and Shiue [Adv. in Appl. Math. 20 (1998), 366–384]. We show that in the case where $\beta =0$ and $\alpha$ and $r$ are nonnegative integers both of our $(p,q)$-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our $(p,q)$-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our $(p,q)$-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

Classification of Six-Point Metrics

Abstract: There are $339$ combinatorial types of generic metrics on six points. They correspond to the $339$ regular triangulations of the second hypersimplex $\Delta(6,2)$, which also has $14$ non-regular triangulations.

On Rainbow Arithmetic Progressions

Abstract: Consider natural numbers $\{1, \cdots, n\}$ colored in three colors We prove that if each color appears on at least $(n+4)/6$ numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors This variation on the theme of Van der Waerden's theorem proves the conjecture of Jungic et al

Random Cayley Graphs are Expanders: a Simple Proof of the Alon–Roichman Theorem

Abstract: We give a simple proof of the Alon–Roichman theorem, which asserts that the Cayley graph obtained by selecting $c_\varepsilon \log |G|$ elements, independently and uniformly at random, from a finite group $G$ has expected second eigenvalue no more than $\varepsilon$; here $c_\varepsilon$ is a constant that depends only on $\varepsilon$. In particular, such a graph is an expander with constant probability. Our new proof has three advantages over the original proof: (i.) it is extremely simple, relying only on the decomposition of the group algebra and tail bounds for operator-valued random variables, (ii.) it shows that the $\log |G|$ term may be replaced with $\log D$, where $D \leq |G|$ is the sum of the dimensions of the irreducible representations of $G$, and (iii.) it establishes the result above with a smaller constant $c_\varepsilon$.

Goldberg-Coxeter Construction for $3$- and $4$-valent Plane Graphs

Abstract: We consider the Goldberg-Coxeter construction $GC_{k,l}(G_0)$ (a generalization of a simplicial subdivision of the dodecahedron considered by Goldberg [Tohoku Mathematical Journal, 43 (1937) 104–108] and Coxeter [A Spectrum of Mathematics, OUP, (1971) 98–107]), which produces a plane graph from any $3$- or $4$-valent plane graph for integer parameters $k,l$. A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face; a central circuit in a $4$-valent plane graph $G$ is a circuit of edges, such that no two consecutive edges belong to the same face. We study the zigzag (or central circuit) structure of the resulting graph using the algebraic formalism of the moving group , the $(k,l)$-product and a finite index subgroup of $SL_2(\Bbb{Z})$, whose elements preserve the above structure. We also study the intersection pattern of zigzags (or central circuits) of $GC_{k,l}(G_0)$ and consider its projections , obtained by removing all but one zigzags (or central circuits).

Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice

Abstract: We show that symmetric Venn diagrams for $n$ sets exist for every prime $n$, settling an open question. Until this time, $n=11$ was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice ${\cal B}_n$, and prove that such decompositions exist for all prime $n$. A consequence of the approach is a constructive proof that the quotient poset of ${\cal B}_n$, under the relation "equivalence under rotation", has a symmetric chain decomposition whenever $n$ is prime. We also show how symmetric chain decompositions can be used to construct, for all $n$, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.

Strings with maximally many distinct subsequences and substrings

Abstract: A natural problem in extremal combinatorics is to maximize the number of distinct subsequences for any length-$n$ string over a finite alphabet $\Sigma$; this value grows exponentially, but slower than $2^n$. We use the probabilistic method to determine the maximizing string, which is a cyclically repeating string. The number of distinct subsequences is exactly enumerated by a generating function, from which we also derive asymptotic estimates. For the alphabet $\Sigma=\{1,2\}$, $\,(1,2,1,2,\dots)$ has the maximum number of distinct subsequences, namely ${\rm Fib}(n+3)-1 \sim \left((1+\sqrt5)/2\right)^{n+3} \! / \sqrt{5}$. We also consider the same problem with sub strings in lieu of sub sequences . Here, we show that an appropriately truncated de Bruijn word attains the maximum. For both problems, we compare the performance of random strings with that of the optimal ones.

On-line Ramsey theory.

Abstract: The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph $H$, keeping the constructed graph in a prescribed class ${\cal G}$. The main problem is to recognize the winner for a given pair $H,{\cal G}$. In particular, we prove that Builder has a winning strategy for any $k$-colorable graph $H$ in the game played on $k$-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for $3$-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.

Airy Phenomena and Analytic Combinatorics of Connected Graphs

Abstract: Until now, the enumeration of connected graphs has been dealt with by probabilistic methods, by special combinatorial decompositions or by somewhat indirect formal series manipulations. We show here that it is possible to make analytic sense of the divergent series that expresses the generating function of connected graphs. As a consequence, it becomes possible to derive analytically known enumeration results using only first principles of combinatorial analysis and straight asymptotic analysis—specifically, the saddle-point method. In this perspective, the enumeration of connected graphs by excess (of number of edges over number of vertices) derives from a simple saddle-point analysis. Furthermore, a refined analysis based on coalescent saddle points yields complete asymptotic expansions for the number of graphs of fixed excess, through an explicit connection with Airy functions.

On the Associative Nijenhuis Relation

Abstract: We give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product.

h-vectors, Eulerian polynomials and stable polytopes of graphs

Abstract: Conditions are given on a lattice polytope $P$ of dimension $m$ or its associated affine semigroup ring which imply inequalities for the $h^*$-vector $(h^*_0, h^*_1,\dots,h^*_m)$ of $P$ of the form $h^*_i \ge h^*_{d-i}$ for $1 \le i \le \lfloor d / 2 \rfloor$ and $h^*_{\lfloor d / 2 \rfloor} \ge h^*_{\lfloor d / 2 \rfloor + 1} \ge \cdots \ge h^*_d$, where $h^*_i = 0$ for $d

Generalizing Narayana and Schroder Numbers to Higher Dimensions

Abstract: Let ${\cal C}(d,n)$ denote the set of $d$-dimensional lattice paths using the steps $X_1 := (1, 0, \ldots, 0),$ $ X_2 := (0, 1, \ldots, 0),$ $\ldots,$ $ X_d := (0,0, \ldots,1)$, running from $(0,\ldots,0)$ to $(n,\ldots,n)$, and lying in $\{(x_1,x_2, \ldots, x_d) : 0 \le x_1 \le x_2 \le \ldots \le x_d \}$. On any path $P:=p_1p_2 \ldots p_{dn} \in {\cal C}(d,n)$, define the statistics ${\rm asc}(P) := $$|\{i : p_ip_{i+1} = X_jX_{\ell}, j \ell \}|$. Define the generalized Narayana number $N(d,n,k)$ to count the paths in ${\cal C}(d,n)$ with ${\rm asc}(P)=k$. We consider the derivation of a formula for $N(d,n,k)$, implicit in MacMahon's work. We examine other statistics for $N(d,n,k)$ and show that the statistics ${\rm asc}$ and ${\rm des}-d+1$ are equidistributed. We use Wegschaider's algorithm, extending Sister Celine's (Wilf-Zeilberger) method to multiple summation, to obtain recurrences for $N(3,n,k)$. We introduce the generalized large Schroder numbers $(2^{d-1}\sum_k N(d,n,k)2^k)_{n\ge1}$ to count constrained paths using step sets which include diagonal steps.

Tight Estimates for Eigenvalues of Regular Graphs

Abstract: It is shown that if a $d$-regular graph contains $s$ vertices so that the distance between any pair is at least $4k$, then its adjacency matrix has at least $s$ eigenvalues which are at least $2 \sqrt {d-1} \cos \big({\pi\over 2 k}\big)$. A similar result has been proved by Friedman using more sophisticated tools.

On Feasible Sets of Mixed Hypergraphs

Abstract: A mixed hypergraph $H$ is a triple $(V,{\cal C},{\cal D})$ where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, called ${\cal C}$-edges and ${\cal D}$-edges. A vertex coloring of $H$ is proper if each ${\cal C}$-edge contains two vertices with the same color and each ${\cal D}$-edge contains two vertices with different colors. The spectrum of $H$ is a vector $(r_1,\ldots,r_m)$ such that there exist exactly $r_i$ different colorings using exactly $i$ colors, $r_m\ge 1$ and there is no coloring using more than $m$ colors. The feasible set of $H$ is the set of all $i$'s such that $r_i e 0$. We construct a mixed hypergraph with $O(\sum_i\log r_i)$ vertices whose spectrum is equal to $(r_1,\ldots,r_m)$ for each vector of non-negative integers with $r_1=0$. We further prove that for any fixed finite sets of positive integers $A_1\subset A_2$ ($1 otin A_2$), it is NP-hard to decide whether the feasible set of a given mixed hypergraph is equal to $A_2$ even if it is promised that it is either $A_1$ or $A_2$. This fact has several interesting corollaries, e.g., that deciding whether a feasible set of a mixed hypergraph is gap-free is both NP-hard and coNP-hard.

On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane

Abstract: Let $G$ be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane. We show that every such graph with clique number $k$ is $(3k-3)$-degenerate. This bound is sharp. As a consequence, we derive that $G$ is $(3k-2)$-colorable. We show also that the chromatic number of every intersection graph $H$ of a family of homothetic copies of a fixed convex set in the plane with clique number $k$ is at most $6k-6$.