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Algebraic study on Cameron-Walker graphs | | Takayuki Hibi
; Akihiro Higashitani
; Kyouko Kimura
; Augustine B. O'Keefe
; | | Date: |
22 Aug 2013 | | Abstract: | Let $G$ be a finite simple graph on $[n]$ and $I(G) subset S$ the edge ideal
of $G$, where $S = K[x_{1}, ldots, x_{n}]$ 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 $im(G) leq
eg(S/I(G)) leq m(G)$,
where $
eg(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
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 every Cameron-Walker graph is
sequentially Cohen-Macaulay. | | Source: | arXiv, 1308.4765 | | Services: | Forum | Review | PDF | Favorites | |
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