Mohammad Farrokhi Derakhshandeh Ghouchan

Let $Γ=(V,E)$ be a finite simple graph. A matching $M \subseteq E$ is positive if there exists a weight function on $V$ such that the matching $M$ is characterized by those edges with positive weights. A positive matching decomposition (pmd) of $Γ$ with $p$ parts is an ordered partition $E_1,\ldots,E_p$ of $E$ such that $E_i$ is a positive matching of $(V, E \setminus \bigcup_{j=1}^{i-1} E_j)$, for $i = 1, \ldots, p$. The smallest $p$ for which $Γ$ admits a pmd with $p$ parts is denoted by $\mathrm{pmd}(Γ)$. We study the pmd of the Cartesian product of graphs and give sharp upper bounds for them in terms of the pmds and chromatic numbers of their components. In special cases, we compute the pmd of grid graphs that is the Cartesian product of paths and cycles.

Let $\mathbb{K}$ be a field and $n$ be a positive integer. Let $Γ=([n], E)$ be a simple graph, where $[n]=\{1,\ldots, n\}$. If $S=\mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ is a polynomial ring, then the graded ideal \[ L_Γ^\mathbb{K}(2) = \left( x_{i}x_{j} + y_{i}y_{j} \colon \quad \{i, j\} \in E(Γ)\right) \subset S,\] is called the Lovász-Saks-Schrijver ideal, LSS-ideal for short, of $Γ$ with respect to $\mathbb{K}$. In the present paper, we compute a Gröbner basis of this ideal with respect to lexicographic ordering induced by $x_1>\cdots>x_n>y_1>\cdots>y_n$ when $Γ=T$ is a tree. As a result, we show that it is independent of the choice of the ground field $\mathbb{K}$ and compute the Hilbert series of $L_T^\mathbb{K}(2)$. Finally, we present concrete combinatorial formulas to obtain the Krull dimension of $S/L_T^\mathbb{K}(2)$ as well as lower and upper bounds for Krull dimension.

Consider an $m\times n$ table $T$ and latices paths $\nu_1,\ldots,\nu_k$ in $T$ such that each step $\nu_{i+1}-\nu_i=(1,1)$, $(1,0)$ or $(1,-1)$. The number of paths from the $(1,i)$-blank (resp. first column) to the $(s,t)$-blank is denoted by $\mathcal{D}^i(s,t)$ (resp. $\mathcal{D}(s,t)$). Also, the number of all paths form the first column to the las column is denoted by $\mathcal{I}_m(n)$. We give explicit formulas for the numbers $\mathcal{D}^1(s,t)$ and $\mathcal{D}(s,t)$.

Let $\mathbb{K}$ be a field and $I$ be a square-free monomial ideal in the polynomial ring $\mathbb{K}[x_1, \ldots, x_n]$. The Green-Lazarsfeld index, $\mathrm{index}(I)$, counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of $I$. In this paper, we study this invariant for $I$ and its powers from a combinatorial point of view. We characterize all square-free monomial ideals $I$ generated in degree $3$ such that $\mathrm{index}(I)>1$. Utilizing this result, we also characterize all square-free monomial ideals generated in degree $3$ such that $\mathrm{index}(I)>1$ and $\mathrm{index}(I^2)=1$. In case $n\leq5$, it is shown that $\mathrm{index}(I^k)>1$ for all $k$ if $I$ is any square-free monomial ideal with $\mathrm{index}(I)>1$.