Algebraic study on Cameron–Walker graphs

• Published in: Journal of algebra 2015-01, Vol.422, p.257-269
• Main Authors: Hibi, Takayuki, Higashitani, Akihiro, Kimura, Kyouko, O'Keefe, Augustine B.
• Format: Article
• Language: English
• Subjects: Cameron–Walker graph; Cohen–Macaulay graph; Edge ideal; Finite graph; Gorenstein graph; Sequentially Cohen–Macaulay graph; Unmixed graph
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2014.07.037

Let G be a finite simple graph on [n] and I(G)⊂S the edge ideal of G, where S=K[x1,…,xn] is the polynomial ring over a field K. Let m(G) denote the maximum size of matchings of G and im(G) that of induced matchings of G. It is known that im(G)≤reg(S/I(G))≤m(G), where reg(S/I(G)) is the Castelnuovo–Mumford regularity of S/I(G). Cameron and Walker succeeded in classifying the finite connected simple graphs G with im(G)=m(G). We say that a finite connected simple graph G is a Cameron–Walker graph if im(G)=m(G) and if G is neither a star nor a star triangle. In the present paper, we study Cameron–Walker graphs from a viewpoint of commutative algebra. First, we prove that a Cameron–Walker graph G is unmixed if and only if G is Cohen–Macaulay and classify all Cohen–Macaulay Cameron–Walker graphs. Second, we prove that there is no Gorenstein Cameron–Walker graph. Finally, we prove that every Cameron–Walker graph is sequentially Cohen–Macaulay.