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26. CMB 2013 (vol 57 pp. 188)

Rad, Nader Jafari; Jafari, Sayyed Heidar
A Characterization of Bipartite Zero-divisor Graphs
In this paper we obtain a characterization for all bipartite zero-divisor graphs of commutative rings $R$ with $1$, such that $R$ is finite or $|Nil(R)|\neq2$.

Keywords:zero-divisor graph, bipartite graph
Categories:13AXX, 05C25

27. CMB 2013 (vol 56 pp. 449)

Akbari, S.; Chavooshi, M.; Ghanbari, M.; Zare, S.
The $f$-Chromatic Index of a Graph Whose $f$-Core has Maximum Degree $2$
Let $G$ be a graph. The minimum number of colors needed to color the edges of $G$ is called the chromatic index of $G$ and is denoted by $\chi'(G)$. It is well-known that $\Delta(G) \leq \chi'(G) \leq \Delta(G)+1$, for any graph $G$, where $\Delta(G)$ denotes the maximum degree of $G$. A graph $G$ is said to be Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if $\chi'(G) = \Delta(G) + 1$. Also, $G_\Delta$ is the induced subgraph on all vertices of degree $\Delta(G)$. Let $f:V(G)\rightarrow \mathbb{N}$ be a function. An $f$-coloring of a graph $G$ is a coloring of the edges of $E(G)$ such that each color appears at each vertex $v\in V(G)$ at most $f (v)$ times. The minimum number of colors needed to $f$-color $G$ is called the $f$-chromatic index of $G$ and is denoted by $\chi'_{f}(G)$. It was shown that for every graph $G$, $\Delta_{f}(G)\le \chi'_{f}(G)\le \Delta_{f}(G)+1$, where $\Delta_{f}(G)=\max_{v\in V(G)} \big\lceil \frac{d_G(v)}{f(v)}\big\rceil$. A graph $G$ is said to be $f$-Class $1$ if $\chi'_{f}(G)=\Delta_{f}(G)$, and $f$-Class $2$, otherwise. Also, $G_{\Delta_f}$ is the induced subgraph of $G$ on $\{v\in V(G):\,\frac{d_G(v)}{f(v)}=\Delta_{f}(G)\}$. Hilton and Zhao showed that if $G_{\Delta}$ has maximum degree two and $G$ is Class $2$, then $G$ is critical, $G_{\Delta}$ is a disjoint union of cycles and $\delta(G)=\Delta(G)-1$, where $\delta(G)$ denotes the minimum degree of $G$, respectively. In this paper, we generalize this theorem to $f$-coloring of graphs. Also, we determine the $f$-chromatic index of a connected graph $G$ with $|G_{\Delta_f}|\le 4$.

Keywords:$f$-coloring, $f$-Core, $f$-Class $1$
Categories:05C15, 05C38

28. CMB 2013 (vol 57 pp. 210)

Zhang, Jiao; Wang, Qing-Wen
An Explicit Formula for the Generalized Cyclic Shuffle Map
We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules. Using this formula we give a combinatorial proof of the generalized cyclic Eilenberg-Zilber theorem.

Keywords:generalized Cyclic shuffle map, Cylindrical module, Eilenberg-Zilber theorem, Cyclic homology
Categories:19D55, 05E45

29. CMB 2012 (vol 56 pp. 709)

Bartošová, Dana
Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
It is a well-known fact, that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G.$ We apply the original description by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using Fraïssé theory and Ramsey theory. This work generalizes some of the known results about countable structures.

Keywords:universal minimal flows, ultrafilter flows, Ramsey theory
Categories:37B05, 03E02, 05D10, 22F50, 54H20

30. CMB 2011 (vol 55 pp. 418)

Vinh, Le Anh
Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields
Given a positive integer $n$, a finite field $\mathbb{F}_q$ of $q$ elements ($q$ odd), and a non-degenerate symmetric bilinear form $B$ on $\mathbb{F}_q^n$, we determine the largest possible cardinality of pairwise $B$-orthogonal subsets $\mathcal{E} \subseteq \mathbb{F}_q^n$, that is, for any two vectors $\mathbf{x}, \mathbf{y} \in \mathcal{E}$, one has $B (\mathbf{x}, \mathbf{y}) = 0$.

Keywords:orthogonal sets, zero-distance sets

31. CMB 2011 (vol 56 pp. 265)

Chen, Yichao; Mansour, Toufik; Zou, Qian
Embedding Distributions of Generalized Fan Graphs
Total embedding distributions have been known for a few classes of graphs. Chen, Gross, and Rieper computed it for necklaces, close-end ladders and cobblestone paths. Kwak and Shim computed it for bouquets of circles and dipoles. In this paper, a splitting theorem is generalized and the embedding distributions of generalized fan graphs are obtained.

Keywords:total embedding distribution, splitting theorem, generalized fan graphs

32. CMB 2011 (vol 56 pp. 407)

Rad, Nader Jafari; Jafari, Sayyed Heidar; Mojdeh, Doost Ali
On Domination in Zero-Divisor Graphs
We first determine the domination number for the zero-divisor graph of the product of two commutative rings with $1$. We then calculate the domination number for the zero-divisor graph of any commutative artinian ring. Finally, we extend some of the results to non-commutative rings in which an element is a left zero-divisor if and only if it is a right zero-divisor.

Keywords:zero-divisor graph, domination
Categories:13AXX, 05C69

33. CMB 2011 (vol 55 pp. 462)

Campbell, Peter S.; Stokke, Anna
Hook-content Formulae for Symplectic and Orthogonal Tableaux
By considering the specialisation $s_{\lambda}(1,q,q^2,\dots,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $\lambda$ in terms of the contents and hook lengths of the boxes in the Young diagram. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $\lambda$-tableaux.

Keywords:symplectic tableaux, orthogonal tableaux, Schur function
Categories:05E05, 05E10

34. CMB 2011 (vol 55 pp. 410)

Service, Robert
A Ramsey Theorem with an Application to Sequences in Banach Spaces
The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of Galvin's theorem is used in the proof. An alternative proof of the dichotomy result for sequences in Banach spaces is also sketched, using the Galvin-Prikry theorem.

Keywords:Banach spaces, Ramsey theory
Categories:46B15, 05D10

35. CMB 2011 (vol 54 pp. 255)

Dehaye, Paul-Olivier
On an Identity due to Bump and Diaconis, and Tracy and Widom
A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener-Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump-Diaconis-Tracy-Widom identity is a differentiated version of the classical Jacobi-Trudi identity.

Keywords:Toeplitz matrices, Jacobi-Trudi identity, Szegő limit theorem, Heine identity, Wiener-Hopf factorization
Categories:47B35, 05E05, 20G05

36. CMB 2011 (vol 54 pp. 217)

Chen, William Y. C.; Wang, Larry X. W.; Yang, Arthur L. B.
Recurrence Relations for Strongly $q$-Log-Convex Polynomials
We consider a class of strongly $q$-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly $q$-log-convex. We also prove that the Bessel transformation preserves log-convexity.

Keywords:log-concavity, $q$-log-convexity, strong $q$-log-convexity, Bell polynomials, Bessel polynomials, Ramanujan polynomials, Dowling polynomials
Categories:05A20, 05E99

37. CMB 2010 (vol 53 pp. 757)

Woo, Alexander
Interval Pattern Avoidance for Arbitrary Root Systems
We extend the idea of interval pattern avoidance defined by Yong and the author for $S_n$ to arbitrary Weyl groups using the definition of pattern avoidance due to Billey and Braden, and Billey and Postnikov. We show that, as previously shown by Yong and the author for $\operatorname{GL}_n$, interval pattern avoidance is a universal tool for characterizing which Schubert varieties have certain local properties, and where these local properties hold.

Categories:14M15, 05E15

38. CMB 2010 (vol 53 pp. 425)

Chapoton, Frédéric
Free Pre-Lie Algebras are Free as Lie Algebras
We prove that the $\mathfrak{S}$-module $\operatorname{PreLie}$ is a free Lie algebra in the category of $\mathfrak{S}$-modules and can therefore be written as the composition of the $\mathfrak{S}$-module $\operatorname{Lie}$ with a new $\mathfrak{S}$-module $X$. This implies that free pre-Lie algebras in the category of vector spaces, when considered as Lie algebras, are free on generators that can be described using $X$. Furthermore, we define a natural filtration on the $\mathfrak{S}$-module $X$. We also obtain a relationship between $X$ and the $\mathfrak{S}$-module coming from the anticyclic structure of the $\operatorname{PreLie}$ operad.

Categories:18D50, 17B01, 18G40, 05C05

39. CMB 2010 (vol 53 pp. 453)

Desgroseilliers, Marc; Larose, Benoit; Malvenuto, Claudia; Vincent, Christelle
Some Results on Two Conjectures of Schützenberger
We present some partial results concerning two conjectures of Schützenberger on evacuations of Young tableaux.

Keywords:Evacuation of Standard Young tableaux
Categories:05E10, 05A99

40. CMB 2009 (vol 53 pp. 3)

Athanasiadis, Christos A.
A Combinatorial Reciprocity Theorem for Hyperplane Arrangements
Given a nonnegative integer $m$ and a finite collection $\mathcal A$ of linear forms on $\mathcal Q^d$, the arrangement of affine hyperplanes in $\mathcal Q^d$ defined by the equations $\alpha(x) = k$ for $\alpha \in \mathcal A$ and integers $k \in [-m, m]$ is denoted by $\mathcal A^m$. It is proved that the coefficients of the characteristic polynomial of $\mathcal A^m$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law.

Categories:52C35, 05E99

41. CMB 2009 (vol 53 pp. 378)

Zhou, Sizhong
A New Sufficient Condition for a Graph To Be $(g,f,n)$-Critical
Let $G$ be a graph of order $p$, let $a$, $b$, and $n$ be nonnegative integers with $1\leq a\lt b$, and let $g$ and $f$ be two integer-valued functions defined on $V(G)$ such that $a\leq g(x)\lt f(x)\leq b$ for all $x\in V(G)$. A $(g,f)$-factor of graph $G$ is a spanning subgraph $F$ of $G$ such that $g(x)\leq d_F(x)\leq f(x)$ for each $x\in V(F)$. Then a graph $G$ is called $(g,f,n)$-critical if after deleting any $n$ vertices of $G$ the remaining graph of $G$ has a $(g,f)$-factor. The binding number $\operatorname{bind}(G)$ of $G$ is the minimum value of ${|N_G(X)|}/{|X|}$ taken over all non-empty subsets $X$ of $V(G)$ such that $N_G(X)\neq V(G)$. In this paper, it is proved that $G$ is a $(g,f,n)$-critical graph if \[ \operatorname{bind}(G)\gt \frac{(a+b-1)(p-1)}{(a+1)p-(a+b)-bn+2} \quad\text{and}\quad p\geq \frac{(a+b-1)(a+b-2)}{a+1}+\frac{bn}{a}. \] Furthermore, it is shown that this result is best possible in some sense.

Keywords:graph, $(g,f)$-factor, $(g,f,n)$-critical graph, binding number

42. CMB 2009 (vol 53 pp. 171)

Thomas, Hugh; Yong, Alexander
Multiplicity-Free Schubert Calculus
Multiplicity-free algebraic geometry is the study of subvarieties $Y\subseteq X$ with the ``smallest invariants'' as witnessed by a multiplicity-free Chow ring decomposition of $[Y]\in A^{\star}(X)$ into a predetermined linear basis. This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.

Categories:14M15, 14M05, 05E99

43. CMB 2009 (vol 52 pp. 451)

Pach, János; Tardos, Gábor; Tóth, Géza
Indecomposable Coverings
We prove that for every $k>1$, there exist $k$-fold coverings of the plane (i) with strips, (ii) with axis-parallel rectangles, and (iii) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct for every $k>1$ a set of points $P$ and a family of disks $\cal D$ in the plane, each containing at least $k$ elements of $P$, such that, no matter how we color the points of $P$ with two colors, there exists a disk $D\in{\cal D}$ all of whose points are of the same color.

Categories:52C15, 05C15

44. CMB 2009 (vol 52 pp. 416)

Malik, Shabnam; Qureshi, Ahmad Mahmood; Zamfirescu, Tudor
Hamiltonian Properties of Generalized Halin Graphs
A Halin graph is a graph $H=T\cup C$, where $T$ is a tree with no vertex of degree two, and $C$ is a cycle connecting the end-vertices of $T$ in the cyclic order determined by a plane embedding of $T$. In this paper, we define classes of generalized Halin graphs, called $k$-Halin graphs, and investigate their Hamiltonian properties.

Keywords:$k$-Halin graph, Hamiltonian, Hamiltonian connected, traceable
Categories:05C45, 05C38

45. CMB 2008 (vol 51 pp. 584)

Purbhoo, Kevin; Willigenburg, Stephanie van
On Tensor Products of Polynomial Representations
We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $\GL(n,\mathbb{C})$ is isomorphic to another. As a consequence we discover families of Littlewood--Richardson coefficients that are non-zero, and a condition on Schur non-negativity.

Keywords:polynomial representation, symmetric function, Littlewood--Richardson coefficient, Schur non-negative
Categories:05E05, 05E10, 20C30

46. CMB 2008 (vol 51 pp. 535)

Csorba, Péter
On the Simple $\Z_2$-homotopy Types of Graph Complexes and Their Simple $\Z_2$-universality
We prove that the neighborhood complex $\N(G)$, the box complex $\B(G)$, the homomorphism complex $\Hom(K_2,G)$and the Lov\'{a}sz complex $\L(G)$ have the same simple $\Z_2$-homotopy type in the sense of Whitehead. We show that these graph complexes are simple $\Z_2$-universal.

Keywords:graph complexes, simple $\Z_2$-homotopy, universality
Categories:57Q10, 05C10, 55P10

47. CMB 2008 (vol 51 pp. 413)

Thé, L. Nguyen Van
Big Ramsey Degrees and Divisibility in Classes of Ultrametric Spaces
Given a countable set $S$ of positive reals, we study finite-dimensional Ramsey-theoretic properties of the countable ultrametric Urysohn space $\textbf{Q} _S$ with distances in $S$.

Keywords:Ramsey theory, Urysohn metric spaces, ultrametric spaces
Categories:05C50, 54E35

48. CMB 2008 (vol 51 pp. 424)

Novelli, Jean-Christophe; Thibon, Jean-Yves
Noncommutative Symmetric Bessel Functions
The consideration of tensor products of $0$-Hecke algebra modules leads to natural analogs of the Bessel $J$-functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions.

Categories:05E05, 16W30, 05A15

49. CMB 2008 (vol 51 pp. 47)

Croot, Ernie
The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit
How few three-term arithmetic progressions can a subset $S \subseteq \Z_N := \Z/N\Z$ have if $|S| \geq \upsilon N$ (that is, $S$ has density at least $\upsilon$)? Varnavides %\cite{varnavides} showed that this number of arithmetic progressions is at least $c(\upsilon)N^2$ for sufficiently large integers $N$. It is well known that determining good lower bounds for $c(\upsilon)> 0$ is at the same level of depth as Erd\" os's famous conjecture about whether a subset $T$ of the naturals where $\sum_{n \in T} 1/n$ diverges, has a $k$-term arithmetic progression for $k=3$ (that is, a three-term arithmetic progression). We answer a question posed by B. Green %\cite{AIM} about how this minimial number of progressions oscillates for a fixed density $\upsilon$ as $N$ runs through the primes, and as $N$ runs through the odd positive integers.


50. CMB 2007 (vol 50 pp. 504)

Dukes, Peter; Ling, Alan C. H.
Asymptotic Existence of Resolvable Graph Designs
Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A \emph{block design} $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda$ elements of $\cA$. More generally, for a fixed simple graph $G$, a \emph{graph design} $\GD(v,G,\lambda)$ is a collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda$ elements of $\cA$. A famous result of Wilson says that for a fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all sufficiently large $v$ satisfying certain necessary conditions. A block (graph) design as above is \emph{resolvable} if $\cA$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

Keywords:graph decomposition, resolvable designs
Categories:05B05, 05C70, 05B10
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