Some algebraic properties of the ideals of Veronese bi-type arising from graphs with loops are studied. More precisely, the property of these ideals to be bi-polymatroidal is discussed. Moreover, we are able to determine the structure of the ideals of vertex covers for such generalized graph ideals. | Turk J Math (2016) 40: 753 – 765 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On algebraic properties of Veronese bi-type ideals arising from graphs Maurizio IMBESI, Monica LA BARBIERA∗ Department of Mathematical and Computer Sciences, Physical Sciences, and Earth Sciences, University of Messina, Messina, Italy Received: • Accepted/Published Online: • Final Version: Abstract: Some algebraic properties of the ideals of Veronese bi-type arising from graphs with loops are studied. More precisely, the property of these ideals to be bi-polymatroidal is discussed. Moreover, we are able to determine the structure of the ideals of vertex covers for such generalized graph ideals. Key words: Veronese bi-type ideals, graph ideals, ideals of vertex covers 1. Introduction Let R = K[X1 , . . . , Xn ; Y1 , . . . , Ym ] be a polynomial ring in two sets of variables over a field K . In some recent papers [5, 9, 16], monomial ideals of R were introduced and their connection to bipartite complete graphs was studied. Here we consider a class of monomial ideals of R , the so-called Veronese bi-type ideals, which are an extension of the ideals of Veronese type in a polynomial ring with two sets of variables. More precisely, the ideals of Veronese bi-type are monomial ideals of R generated in the same degree: Lq,s = ∑ k+r=q Ik,s Jr,s , with k, r ⩾ 1, s ⩽ q , where Ik,s is the Veronese type ideal generated on degree k by the set ∑n a a {X1 i1 · · · Xnin | j=1 aij = k, 0 ⩽ aij ⩽ s, s ∈ {1, . . . , k}} and Jr,s is the Veronese type ideal generated on ∑m b b degree r by the set {Y1 i1 · · · Ymim | j=1 bij = r, 0 ⩽ bij ⩽ s, s ∈ {1, . . . , r}} [10–13, 15]. When s = 2, the Veronese bi-type ideals arise from bipartite graphs with loops, the so-called strong quasi-bipartite graphs [10]. A graph G with loops is quasi-bipartite if its vertex set V can be partitioned into .
