The Strong Shellability Of χ-complexes And Their Generalizations

Giving a complex Δ and a s-coloring χ of Δ,a class of simplicial complex Δχcan be defined,called the χ-complex of Δ.In[5](Biermann et al.,2013),it is proved that χ-complexes are shellable pure complexes,and a general shelling order is given.The strong shellability,as a kind of shellability,is defined by imposing stronger requirements on shelling order,see[17](Guo et al.,2019)for details.In this paper,we mainly study the strong shellability of χ-complexes and their generalizations.Firstly,we construct an injective map between the facet set of Δχ and the set of vectors.Based on the lexicographic order of vectors,we define a total order>x on the facet set of Δχ.We prove that if Δ is a TA complexthen for any coloring χ,>χ is a strong shelling order,hence χ-complex Δχ is a strongly shellable complex.Then we prove that the χ-complex of the complex Δ under any coloring χ is matroid if and only if Δ is a simplex.Secondly,as generalizations of the χ-complex,two classes of new complexes denoted by Δχc and Δχ|c| are studied.Similarly,by the lexicographic order of vectors,we define a total order(?)on the facet set of Δχc and a total order>χ|c| on the facet set of Δχ|x|respectively.Therefore,it is proved that both Δχc and Δχ|c| have the strong shellability if they are TA complex.Finally,we introduce and study the corresponding combinatorial commutative algebraic properties of the graph which are related to Δχc and Δχ|c| respectively.One of the most important results is that Gπc is Cohen-Macaulay if G is a TA graph.