On regularity of symbolic Rees algebras and symbolic powers of vertex cover ideals of graphs

Abstract

In this work, we study the bigraded regularities of the symbolic Rees algebras R s ( J ( G ) ) , R s ( I ( G ) ) R_s(J(G)), R_s(I(G)) , of the vertex cover ideal J ( G ) J(G) and the edge ideal I ( G ) I(G) , of a graph G G respectively. We give combinatorial upper bounds for the ( 1 , 0 ) (1,0) -regularities of R s ( J ( G ) ) R_s(J(G)) and R s ( I ( G ) ) R_s(I(G)) . By using this upper bounds, we give general linear upper bounds for r e g ( J ( G ) ( k ) ) , r e g ( I ( G ) ( k ) ) reg(J(G)^{(k)}), reg(I(G)^{(k)}) for any k ≥ 1 k\geq 1 . Let G G be a graph on n n vertices and deg ⁡ ( J ( G ) ) \deg (J(G)) be the maximum degree of minimal generators of J ( G ) J(G) . We show that if G G is a non-bipartite graph, then k deg ⁡ ( J ( G ) ) ≤ r e g ( J ( G ) ( k ) ) ≤ k deg ⁡ ( J ( G ) ) + α 0 ( G ) − 1 + | A 0 ∪ { x i 1 , … , x i r } | − r , \begin{equation*} k \deg (J(G)) \!\leq \! reg(J(G)^{(k)})\!\leq \! k \deg (J(G))+ \alpha _0(G)-1+|A_0\cup \{x_{i_1}, \ldots , x_{i_r}\}|-r, \end{equation*} for all k ≥ 1 k \geq 1 , where α 0 ( G ) \alpha _0(G) denotes the vertex cover number of G G , A 0 A_0 is a maximal independent set in G G of maximal cardinality, and r r is the number of 0 0 -covers that are present in an irreducible representation of the affine cone associated with the irreducible covers of G G . Also if G G is a non-bipartite perfect graph, then 2 k ≤ r e g ( I ( G ) ( k ) ) ≤ 2 k + n − r + 1 , \begin{equation*} 2k \leq reg(I(G)^{(k)})\leq 2k+n-r+1, \end{equation*} for all k ≥ 1 k \geq 1 , where r r is the number of 0 0 -covers of Γ ( G ) \Gamma (G) that are present in an irreducible representation of the affine cone associated with the irreducible covers of Γ ( G ) \Gamma (G) .