a b s t r a c tA graph is called very well-covered if it is unmixed without isolated vertices such that the cardinality of each minimal vertex cover is half the number of vertices. We first prove that a very well-covered graph is Cohen-Macaulay if and only if it is vertex decomposable. Next, we show that the Castelnuovo-Mumford regularity of the quotient ring of the edge ideal of a very well-covered graph is equal to the maximum number of pairwise 3-disjoint edges.

Available online xxxx Communicated by Louis Rowen MSC: 16W10 16W25 16K20 11E39 Keywords: Central simple algebra Involution Quaternion algebra Frobenius algebra Bilinear and quadratic form Pfister formA necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogonal type.

An exact octagon of Witt groups of central simple algebras with involution is constructed, extending an exact sequence of Parimala, Sridharan and Suresh and motivated by exact sequences obtained by Lewis. From this, we derive relations between the cardinality of certain Witt groups. An exact octagon of equivariant Witt groups is also obtained, thus generalizing a similar octagon constructed by Lewis for quaternion algebras.

Abstract. Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated.