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1. CMB Online first
On Stanley Depths of Certain Monomial Factor Algebras Let $S=K[x_1,\ldots,x_n]$
be the polynomial
ring in $n$-variables over a field $K$ and $I$ a monomial ideal
of $S$. According to one standard primary decomposition of $I$,
we get a Stanley decomposition of the monomial factor algebra
$S/I$.
Using this Stanley decomposition, one can estimate the Stanley
depth of $S/I$. It is proved that
${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {size}}_S(I)$. When $I$ is squarefree
and ${\operatorname {bigsize}}_S(I)\leq 2$, the Stanley conjecture holds
for
$S/I$, i.e., ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {depth}}_S(S/I)$.
Keywords:monomial ideal, size, Stanley depth Categories:13F20, 13C15 |
2. CMB 2014 (vol 58 pp. 105)
On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups |
On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups The prime vertex graph, $\Delta (X)$, and the common divisor graph,
$\Gamma (X)$, are two graphs that have been defined on a set of
positive integers $X$.
Some
properties of these graphs have been studied in the cases where either
$X$ is the set of character degrees of a group or $X$ is the set of
conjugacy class sizes of a group. In this paper, we gather some
results on these graphs arising in the context of direct product of
two groups.
Keywords:prime vertex graph, common divisor graph, character degree, class sizes, graph operation Categories:20E45, 05C25, 05C76 |
3. CMB 2013 (vol 57 pp. 141)
Size, Order, and Connected Domination We give a sharp upper bound on the size of a
triangle-free graph of a given order and connected domination. Our
bound, apart from
strengthening an old classical theorem of Mantel and of
TurÃ¡n , improves on a theorem of Sanchis.
Further, as corollaries, we settle a long standing
conjecture of Graffiti on the leaf number and local independence for
triangle-free graphs and answer a question of Griggs, Kleitman and
Shastri on a lower bound of the leaf number in
triangle-free graphs.
Keywords:size, connected domination, local independence number, leaf number Category:05C69 |