One kind of construction on sunflower with two petals

A sunflower (or ∆-system) with k petals and a core Y is a collection of sets S1,⋯, Sk such that Si∩Sj=Y for all i≠j; the sets S1\Y,⋯, Sk\ Y, are petals. In this paper, we first give a sufficient condition for the existence of a sunflower with 2 petals. Let F={A,B,C} be a family of subsets of a set { a1,⋯,am , b1,⋯,bn , c1,⋯,cn } with ∑ ai m i=1 = ∑ bp n p=1 + ∑ cq n q=1 and A={a1,⋯,am}, B={ b1,⋯,bn } and C={ c1,⋯,cn } are non-increasing lists of nonnegative integers. Suppose that for each r with 1 ≤ r ≤ m, ∑ ai r i=1 ≤ ∑ min{bp, r} n p=1 + ∑ min{cq, r} n q=1 , then the family F * contains a sunflower with two petals, where F * ={G1 ,G2}, G1=G[Y∪X] and G2=[ Z∪X] are the subgraphs induced respectively by Y∪X and Z∪X with dG1(vj) = bj for all vj ∈Y∪X and dG2(vj) = cjfor all vj∈Z∪X. Moreover, we generalize the consequence to the case of a much more general result.


INTRODUCTION
A non-increasing sequence π=(d 1 ,⋯,d n ) of nonnegative integers is said to be graphic if it is the degree sequence of a graph G on n vertices and G is called a realization of π. Many characterizations of graphic lists are known, of which one of the best explicit characterizations is that by Erdos and Gallai, (1960). There have been several proofs of it, including a short constructive proof in Garg et al. (2011).
A k-partite graph is one whose vertex set can be partitioned into k subsets so that no edge has both ends in any one subset. In particular, 2-partite graph and 3partite graph are also called bipartite graph and tripartite graph respectively.
Let A = (a 1 , ⋯, a m ) and B = ( b 1 , ⋯, b n ) be two nonincreasing sequences of nonnegative integers. The pair S=(A;B) is said to be bigraphic if there exists a simple bipartite sets X={x 1 , ⋯, x m } and Y={y 1 , ⋯, y n } such that d G (x i )=a i for 1≤i≤m and d G (y i )=b i for 1≤i≤n. In this case, G is referred to as a realization of S. A well-known theorem due to Gale (1957) and Ryser (1957) independently gives a characterization of S that is bigraphic. In Gale (1957), Tripathi et al. (2010) generalized that theorem and provided a good characterization that is bigraphic on lists of intervals. A sunflower (or ∆-system) with k petals and a core Y is a collection of sets S 1 ,⋯,S k such that S i ∩S j =Y for all i≠j; the sets S 1 \Y ,⋯, S k \Y are k petals and we require that none of them is empty. About sunflower, Erdos and Rado *Corresponding author. E-mail: 158238102@qq.com, 501168265@qq.com #Mathematics Subject Classification (2000): 05C07, 05C70.
Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License discovered the so-called Sunflower Lemma ( Erdos and Rado, 1960) . In this paper, we present a special tripartite graph, which reduces to a sunflower with two petals and a core. Prior to our work, only the degree sequences of bipartite graph were characterized.
A set system or a family of F is a collection of sets. Because of their intimate conceptual relation to graphs, a set system is often called a hypergraph. A family is kuniform if all its members are k-element sets. Thus graphs are k-uniform families with k=2.

THEOREM 1
Let F={A,B,C}be a family of subsets of a set { a 1 ,⋯,a m , b 1 ,⋯,b n , c 1 ,⋯,c n } with ∑ =1 = ∑ =1 + ∑ =1 and A={a 1 ,⋯,a m }, B={b 1 ,⋯,b n } and C={c 1 ,⋯,c n } are nonincreasing lists of nonnegative integers. Suppose that for each r with 1 ≤ ≤ , then the family F * contains a sunflower with two petals, where F * ={ G 1 ,G 2 }, G 1 =G[Y∪X] and G 2 =[ Z∪X] are the subgraphs induced respectively by Y∪X and Z∪X with d G 1 (v j ) = b j for all v j ∈Y∪X and d G 2 (v j ) = b j for all v j ∈Z∪X.
The consequence of Theorem 1 can be generalized to the case of a much more general result.

THEOREM 2
Theorem 2 can be proved by induction on n(n≥3) and the proof technique closely follows that of Theorem 1. So a detailed proof will not be given here.

Proof of Theorem 1
For convenience, let X={x 1 ,⋯,x m } ,Y={ y 1 ,⋯,y n } and Z={ z 1 ,⋯,z n } be three sets of vertices. We shall construct a special tripartite graph G, which will yield the desired sunflower. In fact, G is a realization of degree sequence π=A∪B∪C with vertex-set X∪Y∪Z, i.e., d(x i )=a i for 1≤i≤m, d(y j )=b j for 1≤j≤n and d(z j )=c j for 1+n≤j≤2n. For convenience, we write Y∪Z=W and maintain that X and W are independent sets. We first construct a graph G' with partite sets X, Y and Z satisfying d(x i )=a i for 1≤i≤m, d(v j )≤b j for 1≤j≤n and d(v j )≤c j for 1+n≤j≤2n. Define the critical index to be the largest index r such that d(x i )=a i for 1≤i<r and d(x r )<a r . We will iteratively remove the deficiency a r -d(x r ) at vertex x r , while maintaining d(x i )=a i for 1≤i<r, d(v j )≤b j for 1≤j≤n and d(v j )≤c j for 1+n≤j≤2n. Let S={x r+1 ,⋯,x m }. Note that there exists a vertex v∈[N(x i )\ N(x r )]∩W for 1≤i<r, since d(x i )=a i ≥a r > d(x r ). To prove the theorem we have to consider two cases depending on the degree of v j ∈Y∪Z and its neighbourhood's intersection with X.
Case 1: Suppose, for some j, v j ↔x k for some k>r and v j ∉N(x l )for some l≤r. If l=r, replace v j x k with v j x r . If l<r, replace vx l 、v j x k with vx r 、v j x l . Case 2: Suppose, for some j, d(v j )<b j or d(v j )<c j and v j ∉N(x l )for some l≤r. If l=r, add the edge v j x r . If l<r, replace vx l with vx r 、v j x l .
If none of the cases above arise, an application of (1) gives: Hence d(x r )≥a r . Furthermore, d(x r )≤a r and thus d(x r )=a r . Increasing r by 1 and applying the similar steps leads to the required graph G' with partite sets X and W(that is, Y and Z) satisfying d(x i )=a i for 1≤i≤m, d(y p )≤b p for 1≤p≤n and d(z q )≤c q for 1+n≤q≤2n. On the other hand, since W is an independent set and ∑ ( ) + ∑ ( ) That is, d(y p )=b p for 1≤p≤n and d(z q )=c q for 1≤q≤n. Hence we construct a tripartite graph G satisfying d(x i )=a i for 1≤i≤m, d(y p )=b p for 1≤p≤n and d(z q )=c q for 1+n≤q≤2n. Now let F*=G, G 1 =G[Y∪X] and G 2 =[ Z∪X] be the subgraphs