PARTITION MATROID ARE NICE

MEYSAM ALISHAHI, HOSSEIN HAJIABOLHASSAN, AND FRED´ ERIC´ MEUNIER

1. Introduction For a , the Kneser hypergraph KG( ) is a graph with vertex set E( ) and two vertices are adjacentH if corresponding edges areH disjoint. Alternation number of aH hyper- graph is a combinatorial parameters defined by the first and second authors [1]. For each x =(xH,x ,...,x ) +, , 0 n, an alternating subsequence of x is a sequence x ,...,x of 1 2 n 2{ } i1 ik nonzero terms of x (i1 <

(KG( )) = n alt( ). • H nH + for each x +, , 0 , if alt(x) alt( ) and x > alt( ), then at least one of (x ) • and (x )2 contains{ some} edge of . H | | H H The notation x stands for the cardinality of x which is the number of nonzero terms of x. As a main goal| | of this note, we are going to prove next theorem.

Theorem 1. Let P1,...,Pm be a partition of [n] and let k, s1,...,sm be positive . Assume that P =2s for every i and that k 2. Let be the hypergraph defined by | i|6 i H V ( )=[n] H [n] 1 E( )= A : A P s for every i . H 2 k | \ i| i ⇢ ✓ ◆ If has at least two disjoint edges, then it is nice. H

1The edges of such a hypergraph are the bases of a truncation of a partition matroid. 1 2. Proof of Theorem 1 Throughout this note, k, r, m, and n are positive integers where r 2 and k 1. Furthermore, suppose that ⇡ =(P1,P2,...,Pm) is a partition of [n] and ~s =(s1,s2,...,sm) is a positive vector. The partition matroid Kneser graph PM(⇡; ~s ; k) is a graph with the vertex set V (⇡; ~s ; k)= A : A [n], A = k, 1 i m; A P s { ✓ | | 8   | \ i| i} and two vertices are adjacent if the corresponding are disjoint, i.e., for A, B V (⇡; ~s ; k), 2 A, B is an edge if A B = ?. Note that V (⇡; ~s ; k) can be considered as the edge set of a hypergraph{ } = (⇡; ~s ;\k)withthevertexset[n]. The hypergraph (⇡; ~s ; k) is called the partition H H H matroid hypergraph. Clearly, PM(⇡; ~s ; k) ⇠= KG( (⇡; ~s ; k)) The partition matroid graphs are introducedH in [1] as a generalization of a family of graph named the fractional multiple of the complete graphs introduced in [4]. However, it should be mentioned that the partition matroid graphs are called the multiple Kneser graphs in [1]. For r,t positive integers m, r, and t, the fractional multiple of the complete graph Km has the vertex set consisting of all r-independent sets in a disjoint union of t copies of Km which two vertices are adjacent if their corresponding subsets are disjoint. Clearly, for n = tm, ⇡ =(P1,P2,...,Pm)where r,t P1 = = Pm = t, and ~s =(1, 1,...,1), the graphs PM(⇡; ~s ; r) and Km are isomorphic. Note | | ··· | | r,t that for m = 1, Km = KG(t, r). The chromatic number of the fractional multiple of the complete r,t ⇠ graph Km was studied in [2, 3] and was determined for even m. However, it was conjectures that r,t for odd m 3, (Km )=m(t r + 1). The chromatic number of partition matroid graphs is determined in [1] which gives an armative answer to the aforementioned conjecture. Moreover, there is a colorful type theorem presented in [1] which implies that the equality of chromatic number and circular chromatic number for any partition matroid graph with even chromatic number. In what follows, we prove that the hypergraph (⇡; ~s ; k) with at least two disjoint edges is nice H provided that Pi =2si, for each 1 i m. Although, the chromatic number of PM(⇡; ~s ; k) ⇠= KG( (⇡; ~s ; k))| is| computed6 in [1], but we recompute the chromatic number with a slightly di↵erent formula.H Let ⇡ =(P1,P2,...,Pm) be a partition of [n] and ~s =(s1,s2,...,sm) be a positive integer. One can readily check the following properties of (⇡; ~s ; k). H The hypergraph (⇡; ~s ; k) has nonempty edge set if and only if m f k, • H i=1 j M⇡

M⇡ = max 2k 2+ ( Pj fj): fj 2k 2 J [m] 8 | |  9 ✓ j J j J < X2 X2 = where f =min 2s , P . i i i : ; In the next lemma,{ | we|} present an upper bound for the chromatic number of PM(⇡; ~s ; k).

Lemma 1. Let k, m,andn be positive integers and ~s =(s1,s2,...,sm) be a positive integer vector. m Also, let ⇡ =(P ,P ,...,P ) be a partition of [n] such that f k. Then (PM(⇡; ~s ; k)) 1 2 m j  i=1 max 1,n M X { ⇡} m Proof. Since f k the vertex set of PM(⇡; ~s ; k)) is not empty. If PM(⇡; ~s ; k)) has no edge, j i=1 then (PM(⇡X; ~s ; k)) = 1 which verifies the desired inequality. Now, we may assume that PM(⇡; ~s ; k) 2 m has some edges. It implies that f 2k and consequently, M

m (1) I(x)= min f , alt(x,P ) = f + alt(x,P ) 2k 2 { j j } j j  j=1 j J1 j J2 X X2 X2 This means that f 2k 2, and according to the definition of M ,wehave j  ⇡ j J1 X2 (2) 2k 2+ ( P f ) M . | j| j  ⇡ j J1 X2 Hence,

(3) 2k 2 M + P f . ⇡ | j| j J J1 J J1 X2 X2 Now we have

(4) 2k 2 M + P + alt(x,P ) I(x) 2k 2. ⇡ | j| j   J J1 J J2 X2 X2 Thus

(5) M = alt(x) P + alt(x,P ) M .  | j| j  ⇡ j J1 j J2 X2 X2 + Since alt(x)=M⇡,wehaveK = k 1 and K = k 1 and moreover we have the equality in the inequalities 1, 2, 3, 4, and 5. Having equality in inequalities 2 and 5 implies that

(6) alt(x,P )=2k 2 f . j j j J2 j J1 2 2 X 4 X For a contradiction, suppose that there is some l J such that f < P and f (P )+f 2 2 l | l| ⇡ i l  i J1 2k 2. It implies that X2 M 2k 2+ ( P f ) ⇡ | j| j j J1 l 2X[{ } = M +(P f ) ⇡ | l| l >M⇡ which is impossible. Therefore, for any l J with f < P ,wehave 2 1 l | l| (7) 2k 2 f M⇡, a contradiction. In view of prior discussion, for any l J , we have one of the following condotions 2 2 I) f = P 2s 1, l | l| l II) f < P and 2k 2 f f 2 2s 2. j | j| j  l  l j J1 X2 Therefore, in view if inequality 6, for any l [m] J ,wehave 2 \ 1

alt(x,P) min P 1, 2k 2 f , 2s 2. l  8| l| j 9  l j J1 < X2 = + Consequently, we have alt (x,Pl)

M. Alishahi, School of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran E-mail address: meysam [email protected]

H. Hajiabolhassan, Department of Mathematical Sciences, Shahid Beheshti University, G.C., P.O. Box 19839-69411,Tehran,Iran E-mail address: [email protected]

F. Meunier, Universite´ Paris Est, CERMICS, 77455 Marne-la-Vallee´ CEDEX, France E-mail address: [email protected]

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