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Graph Colorings and Related Symmetric Functions Graph Colorings And Related Symmetric Functions Ideas and Applications Subtitle A Description of Results Interesting Applications Notable Op en problems Richard P Stanley Department of Mathematics Massachusetts Institute of Technology Cambridge MA email rstanmathmitedu version of June 1 Partially supp orted by NSF grant DMS Schur p ositivity Let G b e a nite graph with no lo ops edges from a vertex to itself or multiple edges In we dened a symmetric function X X x x which G G generalizes the chromatic p olynomial n of G In this pap er we will rep ort G on further work related to this symmetric function We rst review the denition of X We will denote by V fv v g G d the vertex set and by E the edge set of G A coloring of G is any function V P f g If is a coloring then set Y x x v v V where x x are commuting indeterminates We say that the coloring is proper if there are no mono chromatic edges ie if uv E then u v Dene X X X x x G G summed over all prop er colorings Thus X is a homogeneous symmet G ric function of degree d in the variables x x x Moreover it is immediate from the denition of X that G n X n G G n where in general for a symmetric function f we denote by f the substi tution x x x x x n n n The basic prop erties of the symmetric function X are discussed in G In particular we considered the expansion of X in terms of the four bases G m the monomial symmetric functions p the p ower sum symmetric func tions s the Schur functions and e the elementary symmetric functions We are assuming a basic knowledge of symmetric functions such as may b e found in Chapter I of One of the most interesting op en problems con cerning X is the following A subp oset Q of a p oset partially ordered set G P is said to b e induced if whenever u v Q and u v in P then u v in Q A nite p oset P is said to b e free if it contains no induced subp oset isomorphic to the disjoint union of a threeelement chain and a oneelement chain We denote the incomparability graph of a p oset P by incP If b is a symmetric function basis then we say that the graph G is bpositive if the expansion of X in the basis b has nonnegative co ecients G Conjecture Conj If P is a free poset then incP is epositive This conjecture is true for free p osets ie the edge complement G of G is bipartite Cor Although the ab ove conjecture remains op en the weaker result that in comparability graphs of free p osets are sp ositive was proved by V Gasharov Ch I I Thm as mentioned in Thm In fact Gasharov gives a combinatorial interpretation of the co ecients which we now explain stated slightly dierently from Gasharov Denition Let P b e a nite p oset with d elements A P tableau of shap e d is a map P P satisfying the following three conditions a For all i we have i i b is a prop er coloring of incP ie if u v then u v or v u c By b the elements of the set i form a chain say u u u Similarly supp ose that the elements of i are v v i v Then for all i and all j we require that v u i j j i+1 Note that if P is itself a chain v v then a map P P d is a P tableau of shap e if and only if the sequence v v is a d lattice p ermutation of shap e as dened eg in p Def Since there is a simple bijection b etween lattice p ermutations of shap e and standard Young tableaux of shap e p we may regard a P tableaux of shap e when P is a chain as a standard Young tableau of shap e Hence for general P a P tableau of shap e should b e regarded as a generalization of a standard Young tableau of shap e Let f P denote the number of P tableaux of shap e Theorem V Gasharov Let P be a free poset and G incP Then X X f P s G d Gasharov proves when P is free by an involution principle argument Since b oth sides have simple combinatorial interpretations there should b e a direct bijective pro of When P is a chain the identity b ecomes X d f s x x x d where f denotes the number of standard Young tableaux of shap e A bi jective pro of of this identity is provided precisely by the RobinsonSchensted corresp ondence so we are seeking a generalization of RobinsonSchensted Such a generalization can b e gleaned from the work of A Magid x though a simpler direct bijection would b e desirable A claw is a complete bipartite graph K A graph is clawfree if no induced subgraph is a claw Note that K is the incomparability graph of the disjoint union of a threeelement chain and oneelement chain and that K is the incomparability graph of no other p oset It follows that an incomparability graph incP is clawfree if and only if P is free Thus it is natural to ask whether Conjecture or Theorem extends to clawfree graphs In Figure we gave an example of a clawfree graph which isnt ep ositive On the other hand the question of whether clawfree graphs might b e sp ositive was rst raised by Gasharov unpublished and there now seems to b e enough evidence to make it into a conjecture Conjecture If G is clawfree then G is spositive There is a nice combinatorial consequence of the sp ositivity of a graph G Recall from that a stable partition of G of type d is a partition of the vertex set V of G into stable or indep endent subsets of sizes Dene the graph G to b e nice if whenever there exists a stable partition of G of type and whenever dominance or ma jorization order called the natural order in p then there exists a stable partition of G of type For instance the claw K is not nice since there exists a stable partition of type but not of type Prop osition If G is spositive then G is nice Pro of By denition of X G p ossesses a stable partition of type if G and only if the co ecient of m in X is nonzero see Prop The G pro of now follows from the fact that the co ecient of m in the Schur function s is nonzero if and only if 2 As a small bit of evidence for Conjecture we have the following result Prop osition A graph G and al l its induced subgraphs are nice if and only if G is clawfree Pro of Since claws are not nice the only if part follows To prove the if part we use the simple fact that if covers in dominance order then is obtained from by subtracting from some part and adding i to some part Not all such need b e covered by Hence it j i suces to prove that if a clawfree graph H has a stable partition of type and if is as just describ ed then H has a stable partition of type Let W b e a subset of V which is the union of a blo ck A of of size and a i blo ck B of size Let H denote the restriction of H to W Hence H is j W W bipartite Since H is clawfree every vertex of H has degree one or two so W H is a disjoint union of paths and cycles The vertices of each path and W cycle alternate b etween A and B Since A B there is a comp onent of H which is a path starting and ending in A Let P denote the vertex set W of this path Replace A and B by A P B P and A P B P This yields a stable partition of H of type completing the pro of 2 Griggs has made a conjecture Problem equivalent to the statement that the incomparability graph of the b o olean algebra B is nice This sug n gests that incB might b e sp ositive which is true for n Perhaps n even the incomparability graph of any distributive lattice is sp ositive This seems quite unlikely however since in particular the distributive lattice L of Figure has the prop erty that incL f g is not sp ositive though incL is itself sp ositive The mo dular lattice of Figure has an incomparability graph which isnt nice and hence isnt sp ositive There is a partition into chains of type but not r r r r r r r r r r r r r r Figure A distributive lattice L for which incL f g isnt sp ositive r r r r r Q Q AA Q A Q Q r r r r r r A AA A A r Figure A mo dular lattice whose incomparability graph isnt nice Ganalogues of symmetric functions For each graph G we dene a homomorphism from the ring of symmet G ric functions to the p olynomial ring in the vertices of G which is closely connected with the symmetric function X This homomorphism is closely G related to and I am grateful to Ira Gessel for calling to my attention the relevance of the pap er Regard the vertices of G as commuting indeterminates and dene for each integer i a p olynomial X Y G e v i S v S where S ranges over all ielement stable subsets of the vertex set V of G In G G particular e We regard e as a Ganalogue of the ith elementary i G symmetric function e Indeed when G has no edges then e e v v i i d i G is not in general a symmetric where V fv v g Note however that e d i function of the vertices of G Let denote the ring of symmetric functions over Z in the variables x x and let ZV denote the p olynomial ring over Z in the vertices of G G Dene a ring homomorphism ZV by setting e e G G i i Since the e s for i are algebraically indep endent and generate i G G is welldened For f we write f f f v and G G G regard f as a Ganalogue of f Closely related to Ganalogues of symmetric functions are certain graphs constructed from G If V N then dene G to b e the graph obtained from G by replacing each vertex v of G by a clique complete subgraph K of size v and placing edges connecting every vertex of K to v v every vertex of K if uv is an edge of G If v then we are simply u
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