We show that the compactness of G-free k-colorability is equivalent to the Boolean prime ideal theorem for any graph G with more than two vertices and any k ≥ 2.

We show that the Castelnuovo–Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices.

Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).

Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).

Let S be a set of n ideals of a commutative ring A and let Geven (respectively Godd) denote the product of all the sums of even (respectively odd) number of ideals of S. If n ≤ 6 the product of Geven and the intersection of all ideals of S is included in Godd. In the case A is an Noetherian integral domain, this inclusion is replaced by equality if and only if A is a Dedekind domain.

The concept of a 0-distributive poset is introduced. It is shown that a section semicomplemented poset is distributive if and only if it is 0-distributive. It is also proved that every pseudocomplemented poset is 0-distributive. Further, 0-distributive posets are characterized in terms of their ideal lattices.

Let I⊆R=K[x1,…,xn] be a toric ideal, i.e., binomial prime ideal. We investigate when the ideal I can “split” into sum of two smaller ideals. For general I, we give sufficient condition for this splitting in terms integer matrix that defines I. When I=IG is finite simple graph G, additional splittings IG related to subgraphs G. there exists I=I1+I2 show some cases describe (multi-)graded Betti n...

Graph burning is a model for spread of information in network. In recent times social networks play major role the thought process people and influences movements, election politics. The number an ideal to calculate speed at which passed throughout burns graph minimum steps. this paper, hybrid algorithm proposed compute network based on selection activators. uses spanning tree, weighted central...

Vizing's conjecture (open since 1968) relates the product of domination numbers two graphs to number their Cartesian graph. In this paper, we formulate as a Positivstellensatz existence question. particular, select classes according vertices and encode an ideal/polynomial pair such that polynomial is non-negative on variety associated with ideal if only true for graph class. Using semidefinite ...

Let F be a binary clutter. We prove that if F is non-ideal, then either F or its blocker b(F) has one of L7,O5,LC7 as a minor. L7 is the non-ideal clutter of the lines of the Fano plane, O5 is the non-ideal clutter of odd circuits of the complete graph K5, and the two-point Fano LC7 is the ideal clutter whose sets are the lines, and their complements, of the Fano plane that contain exactly one ...