26. 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 wellknown 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 

27. CMB 2013 (vol 57 pp. 210)
28. CMB 2012 (vol 56 pp. 709)
 Bartošová, Dana

Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
It is a wellknown 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 FraÃ¯ssÃ© 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 

29. 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 nondegenerate 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, zerodistance sets Category:05B25 

30. 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, closeend 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 Category:05C10 

31. CMB 2011 (vol 56 pp. 407)
 Rad, Nader Jafari; Jafari, Sayyed Heidar; Mojdeh, Doost Ali

On Domination in ZeroDivisor Graphs
We first determine the domination number for the zerodivisor
graph of the product of two commutative rings with $1$. We then
calculate the domination number for the zerodivisor graph of any
commutative artinian ring. Finally, we extend some of the results
to noncommutative rings in which an element is a left
zerodivisor if and only if it is a right zerodivisor.
Keywords:zerodivisor graph, domination Categories:13AXX, 05C69 

32. CMB 2011 (vol 55 pp. 462)
 Campbell, Peter S.; Stokke, Anna

Hookcontent Formulae for Symplectic and Orthogonal Tableaux
By considering the specialisation
$s_{\lambda}(1,q,q^2,\dots,q^{n1})$ 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 

33. 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 GalvinPrikry theorem.
Keywords:Banach spaces, Ramsey theory Categories:46B15, 05D10 

34. CMB 2011 (vol 54 pp. 255)
 Dehaye, PaulOlivier

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 WienerHopf 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
BumpDiaconisTracyWidom identity is a differentiated version of the
classical JacobiTrudi identity.
Keywords:Toeplitz matrices, JacobiTrudi identity, SzegÅ limit theorem, Heine identity, WienerHopf factorization Categories:47B35, 05E05, 20G05 

35. CMB 2011 (vol 54 pp. 217)
 Chen, William Y. C.; Wang, Larry X. W.; Yang, Arthur L. B.

Recurrence Relations for Strongly $q$LogConvex Polynomials
We consider a class of
strongly $q$logconvex 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 $q$logconvex. We also prove
that the Bessel transformation preserves logconvexity.
Keywords:logconcavity, $q$logconvexity, strong $q$logconvexity, Bell polynomials, Bessel polynomials, Ramanujan polynomials, Dowling polynomials Categories:05A20, 05E99 

36. 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 

37. CMB 2010 (vol 53 pp. 425)
 Chapoton, Frédéric

Free PreLie 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 preLie 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 

38. CMB 2010 (vol 53 pp. 453)
39. 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
quasipolynomials in $m$ and that they satisfy a simple combinatorial
reciprocity law.
Categories:52C35, 05E99 

40. 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 integervalued 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 nonempty 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
\[
\operatorname{bind}(G)\gt \frac{(a+b1)(p1)}{(a+1)p(a+b)bn+2}
\quad\text{and}\quad p\geq
\frac{(a+b1)(a+b2)}{a+1}+\frac{bn}{a}.
\]
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 Category:05C70 

41. CMB 2009 (vol 53 pp. 171)
 Thomas, Hugh; Yong, Alexander

MultiplicityFree Schubert Calculus
Multiplicityfree algebraic geometry is the study of subvarieties
$Y\subseteq X$ with the ``smallest invariants'' as witnessed by a
multiplicityfree 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 multiplicityfree Richardson varieties, i.e.,
we classify multiplicityfree products of Schubert classes. This answers
a question of W. Fulton.
Categories:14M15, 14M05, 05E99 

42. 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 axisparallel 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 

43. 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 endvertices
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 

44. 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 LittlewoodRichardson
coefficients that are nonzero, and a condition on Schur nonnegativity.
Keywords:polynomial representation, symmetric function, LittlewoodRichardson coefficient, Schur nonnegative Categories:05E05, 05E10, 20C30 

45. CMB 2008 (vol 51 pp. 535)
46. CMB 2008 (vol 51 pp. 413)
47. CMB 2008 (vol 51 pp. 424)
48. CMB 2008 (vol 51 pp. 47)
 Croot, Ernie

The Minimal Number of ThreeTerm Arithmetic Progressions Modulo a Prime Converges to a Limit
How few threeterm 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 threeterm 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.
Category:05D99 

49. 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 

50. CMB 2007 (vol 50 pp. 632)
 Zelenyuk, Yevhen; Zelenyuk, Yuliya

Transformations and Colorings of Groups
Let $G$ be a compact topological group and let $f\colon G\to G$ be a
continuous transformation of $G$. Define $f^*\colon G\to G$ by
$f^*(x)=f(x^{1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume
that $H=\Imag f^*$ is a subgroup of $G$ and for every
measurable $C\subseteq H$,
$\mu_G((f^*)^{1}(C))=\mu_H(C)$. Then for every measurable
$C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that
$f(Sg^{1})\subseteq Cg^{1}$ and $\mu(S)\ge(\mu(C))^2$.
Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10 
