For every simple graph G,a class of multiple clique cluster-whiskered graphs G^(eπm)is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex Ind G^(eπm)is s...For every simple graph G,a class of multiple clique cluster-whiskered graphs G^(eπm)is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex Ind G^(eπm)is sequentially Cohen-Macaulay.The properties of the graphs G^(eπm)and G^(π)constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind G^(π)and the calculation of Betti numbers of the cover ideal Ic(G^(eπm).We also prove that the complex△=IndH is strongly shellable and pure for either a Boolean graph H=Bn or the full clique-whiskered graph H=G^(W)of C,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△)and the cover ideal Ic(H)have linear quotients.展开更多
We prove that if G is a gap-free and chair-free simple graph,then the regularity of the edge ideal of G is no more than 3.If G is a gap-free and P4-free graph,then it is a chair-free graph;furthermore,the complement o...We prove that if G is a gap-free and chair-free simple graph,then the regularity of the edge ideal of G is no more than 3.If G is a gap-free and P4-free graph,then it is a chair-free graph;furthermore,the complement of G is chordal,and thus the regularity of G is 2.展开更多
基金Supported by the Natural Science Foundation of Shanghai(No.19ZR1424100)the National Natural Science Foundation of China(No.11271250,11971338).
文摘For every simple graph G,a class of multiple clique cluster-whiskered graphs G^(eπm)is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex Ind G^(eπm)is sequentially Cohen-Macaulay.The properties of the graphs G^(eπm)and G^(π)constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind G^(π)and the calculation of Betti numbers of the cover ideal Ic(G^(eπm).We also prove that the complex△=IndH is strongly shellable and pure for either a Boolean graph H=Bn or the full clique-whiskered graph H=G^(W)of C,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△)and the cover ideal Ic(H)have linear quotients.
基金Research supported by the Natural Science Foundation of Shanghai(No.19ZR1424100)the National Natural Science Foundation of China(No.11971338).
文摘We prove that if G is a gap-free and chair-free simple graph,then the regularity of the edge ideal of G is no more than 3.If G is a gap-free and P4-free graph,then it is a chair-free graph;furthermore,the complement of G is chordal,and thus the regularity of G is 2.
