Tur\'an Density of $2$-edge-colored Bipartite Graphs with Application on $\{2, 3\}$-Hypergraphs

We consider the Tur\'an problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$ and $E_b$ do not have to be disjoint. The Tur\'an density $\pi(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/{n\choose 2}$ is the formula to measure the edge density of $G_n$. We will determine the Tur\'an densities of all $2$-edge-colored bipartite graphs. We also give an important application of our study on the Tur\'an problems of $\{2, 3\}$-hypergraphs.


Introduction
Given a graph H, the Turán problem asks for the maximum possible number of edges (denoted as ex(H, n)) in a graph G on n vertices without a copy of H as a sub-graph. The Mantel's theorem [15] states that any graph on n vertices with no triangle contains at most ⌊n 2 /4⌋ edges. Turán [10] proved that the maximal number of edges in a k-clique free graph on n vertices is at most (k − 2)n 2 /(2k − 2). The famed Erdős-Stone-Simonovits Theorem [8,9] proved that the Turán density of any graph H is π(H) = 1 − 1 X (H)−1 , where X (H) is the chromatic number of H. For hypergraphs the extremal problems are harder, see Keevash [14] for a complete survey of some results and methods on uniform hypergraphs. Although Turán type problems for graphs and hypergraphs have been actively studied for decades, there are only few results on non-uniform hypergraphs, see [5,13,11] for related work. Motivated by the study of nonuniform Turán problems [4], in this paper we study a Turán-type problem on the edge-colored graphs and show an application on Turán problems of non-uniform hypergraphs of edge size 2 or 3.
A hypergraph H = (V, E) consists of a vertex set V and an edge set E ⊆ 2 V . An r-uniform hypergraph is a hypergraph such that all its hyperedges have size r. Given positive integers k ≥ r ≥ 2, and a set of colors C, with |C| = k, a k-edge-colored r-uniform hypergraph H (for short, k-colored r-graph) is an r-uniform hypergraph that allows k different colors on each hyperedge. We express H as H = (V, E 1 , E 2 , . . . , E k ) where E i denotes the set of hyperedges colored by ith color in C, note E 1 , E 2 , . . . , E k do not have to be disjoint. We say H ′ is a sub-graph of H, denoted by for every i. Given a family of k-colored r-graphs H, we say G is H-free if it doesn't contain any member of H as a sub-graph. To measure the edge density of G of size n, we use h n (G), which is defined by where the maximum is taken over all H-free k-colored r-graphs G n on n vertices.
By a simple average argument of Katona-Nemetz-Simonovits theorem [12], this limit always exists. where ex(H, n) is the maximum number of hyperedges in an n-vertex H-free k-colored r-graph.
In this paper, we let k = 2. A 2-edge-colored graph is a simple graph (without loops) where each edge is colored either red or blue, or both. We call an edge a double-colored edge if it is colored with both colors. For short, we call the 2-edge-colored graphs simply as 2-colored graphs. A 2-colored graph H can be written as a triple 2 is the set of red edges and E b ⊆ V 2 is the set of blue edges. Denote |E r | and |E b | as the size of each set, denote H r , H b as the induced sub-graphs of H generated by all the red edges and all the blue edges respectively. A graph can be considered as a special 2-colored graph with only one color. We say H is proper if there exists at least one edge in each class E r and E b . Throughout the paper, we consider the proper 2-colored graphs. The results in this paper is finished in year 2018 and recently we were noticed that our study is similar but different to a Turán problem on edge-colored graphs defined by Diwan and Mubayi[1] in which the authors ask for the minimum m, such that the 2-colored graph G, if both its red and blue edges are at least m + 1, contains a given 2-colored graph F ? What we do differently in this paper is the study of the Turán density define above for 2-colored graphs.
It is easy to see that π(H) ≥ 1 for any proper 2-colored graph H, since we can take a complete graph with all edges a single color that does not contain a copy of H.
is not a bipartite graph, one may consider the union of the red complete graph and an extremal graph respect to H b , then the resulting graph is a H-free 2-colored graph with edge density at least 1 + π(H b ) > 1, a contradiction.
In this paper, we will determine the Turán densities of all 2-colored bipartite graphs and characterize the 2-colored graphs achieving these Turán values. The notation [n] is the set of {1, . . . , n}. For convenience, we represent an edge {a, b} by ab.
Definition 2. Given two k-colored r-graphs G and H, a graph homomporhism is a map f : Our consideration on 2-colored graphs is motivated by the study of Turán density of non-uniform hypergraphs, which was first introduced by Johnston and Lu [11], then studied by us [4]. We refer a non-uniform hypergraph H as R-graph, where R is the set of all the cardinalities of edges in H. A degenerate R-graph H has the smallest Turán density, |R| − 1, where |R| is the size of set R. For a history of degenerate extremal graph problems, see [16]. Let r ≥ 3, for r-uniform hypergraphs the r-partite hypergraphs are degenerate and they generalize the bipartite graphs. An interesting problem is what the degenerate non-uniform hypergraph look like? In [4], we prove that except for the case R = {1, 2}, there always exist non-trivial degenerate R-graphs for any set R of two distinct positive integers. The degenerate {1, 3}-graphs are characterized in [4], what about the the degenerate {2, 3}-graphs? In the last section of this paper, we will apply the 2-colored graphs to bound the Turán density of some {2, 3}-graphs.
The paper is organized as follows: in section 2, we show some lemmas on the k-colored r-uniform hypergraphs; in section 3, we classify the Turán densities of all 2-colored bipartite graphs; in section 4, we give an application of the Turán density of 2-colored graphs on {2, 3}-graphs.
2 Lemmas on k-colored r-graphs

Supersaturation and Blowing-up
In this section, we give some definitions and lemmas related to the k-colored r-graphs for k ≥ r ≥ 2. These are natural generalizations from the Turán theory of graphs. We first define the blow-up of a k-colored r-graph.
Lemma 1 (Supersaturation). For any k-colored r-graph H and a > 0, then there are b, n 0 > 0 so that if G is a k-colored r-graph on n > n 0 vertices with h n (G) > π(H) + a then G contains at least b n v(H) copies of H.
Proof. Since we have lim n→∞ π n (H) = π(H), there exists an n 0 > 0 so that if t > n 0 then π t (H) < π(H) + a r . Suppose n > t, and G is a k-colored rgraph on n vertices with h n (G) > π(H) + a. Let T represent any t-set, then G must contain at least a But we also have A contradiction. Since t > n 0 , it follows that each of the a The 'blow-up' does not change the Turán density of k-colored r-graphs.
The following result and proof are natural generalization of results on uniform hypergraphs, see [14].
By the supersaturation lemma, for any a > 0, there are b, n 0 > 0 so that if G is a k-colored r-graph on n > n 0 vertices with h n (G) > π(H) + a then G contains at least b n v(H) copies of H. Consider an auxiliary v(H)-graph U on the same vertex set as G such that the edges of U correspond to copies of H in G. Note that U contains at least b n v(H) edges. For any S > 0, if n is large enough we can find a copy K of K v(H) v(H) (S) in U . Note that K is the complete v(H)-partite v(H)-graph with S vertices in each part, then π(K) = 0. Fix one such K in U . Color each edge of K with one of the v(H)! colors corresponding to the possible orderings with which the vertices of H are mapped into the parts of K. By Ramsey theory, one of the color classes contains at least Note when we say G is H-colorable, it is equivalent to say G is a sub-graph of a blow-up of H. It is easy to prove the following lemmas.
Definition 2. Given two k-colored r-graphs G 1 and G 2 with vertices set V 1 and V 2 , we define the product of G 1 and G 2 , denoted by where e × f is defined through the following way: Proof. There exist two graph homomorphisms f 1 : Define a map f : Thus the map f is a graph homomorphism. Hence G is (G 1 × G 2 )-colorable.

Construction of 2-colored graphs
To compute the lower bound of π(H), we need to construct a family of H-free 2-colored graphs G n with h n (G n ) as large as possible. Here are three useful constructions.
G A : A 2-colored graph G A on n vertices is generated by partitioning the vertex set into two parts such that V (G A ) = X ∪Y and the red edges either meet two vertices in X or meet one vertex in X plus the other in Y , the blue edges meet one vertex in X plus the other in Y . In other words, the red edges which reaches the maximum 4 3 at x = 2 3 .
G B : It is obtained from G A by simply exchanging red edges with blue edges. In other words, the red edges G C : A 2-colored graph G C on n vertices is generated by partitioning the vertex set into two parts such that V (G C ) = A ∪ B and the red edges either meet two vertices in A or meet one vertex in A plus the other in B, the blue edges either meet two vertices in B or meet one vertex in A plus the other in B. In other words, the red edges G D and G E : Two variations of G C are the following constructions: , the red edges {12, 13, 34} and the blue edges {12, 23, 34}: We define a map f : V (H) → {1, 2, 3, 4} as follows: One can check f is a graph homomorphism from the product G A × G B to T .

Turán density of bipartite 2-colored graphs
In this section, we will prove results in Theorem 2. We first give a boundary to divide the Turán densities of 2-colored non-bipartite graphs and 2-colored bipartite graphs.
In particular, π(K 3 ) = 3 2 . Proof. Observe that K 3 is not contained in G C , thus π(K 3 ) ≥ 3 2 . Now we prove the other direction. Let n be a positive integer and G be any K 3 -free 2-colored graph on n vertices. Construct an auxillary graph F on the same vertex set V (G) and with the edge sets consisting of all double-colored edges in G. Let H = E r (F ) consisting of all red colored edges of F . Notice that H is triangle-free. By Mantel theorem, we have Note that H is a subgraph of G and the number of the rest of edges in G is at most n 2 . Therefore, we have This implies that π(K 3 ) = 3 2 . and ex(3, T n ) = 6. In particular, we have π(T 1 ) = 3 2 . Proof. When n ≤ 3, the complete 2-colored graph does not contain T 1 . Thus ex(n, T 1 ) = 0, 0, 2, 6 when n = 0, 1, 2, 3, respectively. The assertion holds for n ≤ 3. It is sufficient to prove for n ≥ 4. Since T 1 is not contained in G C , we have Now we prove the other direction by induction. We may assume n ≥ 4. Let n be a positive integer and G be any T 1 -free 2-colored graph on n vertices.
Then the number of edges in G is: if n = 6, The induction step is finished. It follows that h n (G) ≤ 3 2 . Therefore, π(T 1 ) = 3 2 .
Proof of Lemma 5. For Item 1, let H be a 2-colored non-bipartite graph, without loss of generality, assume H contains an odd cycle with red edges. For any n, let G be a 2-colored graph generated by construction G D , then H can not be contained in G. Similarly, H contains an odd cycle with blue edges, then it is not contained in any 2-colored graph generated by construction G E . Thus π(H) ≥ 3 2 . For Item 2, it is sufficient to prove that any 2-colored bipartite graph H is T 1 -colorable. For any 2-colored bipartite graph H, the sub-graph H r can be partitioned into two disjoint parts V 1 (H r ) and V 2 (H r ) such that the red edges form a bipartite graph between V 1 (H r ) and V 2 (H r ). Similarly for the subgraph H b , the blue edges form a bipartite graph between V 1 (H b ) and V 2 (H b ). Let S be the set of vertices incidents to double colored edges, then S can be divided into four classes: We define a map f : V (H) → {1, 2, 3, 4} as follows: One can verify that this map f is a graph homomorphism from H to T 1 . Hence, π(H) ≤ 3 2 .

The degenerate 2-colored graphs
In this part, we will determine the degenerate 2-colored graphs. Recall the Example 1 in Section 2, we will show that the 2-colored bipartite graph T = ( Proof. We will prove this lemma by induction on n. It is trivial for n = 1, 2, 3, 4. Assume n ≥ 5. We assume that the statement holds for any T -free 2-colored graphs on less than n vertices. Let G = (V, E r , E b ) be a T -free 2-colored graph on n vertices. We also assume G contains at least one double-colored edge uv, or else Then G is one of the following cases.  edges. Note that all edges from X to V 2 are single colored and the number of edges from {u, v} to each vertex in V 2 is at most 2. Thus the total number of edges from V 1 to V 2 is at most |V 1 ||V 2 | edges. Combining these facts together, we have G has at most N edges, where We finish the inductive step. Then we have implying π(T ) = 1. T is degenerate.

Proof of Item 1 of Theorem 2.
Assume H is a degenerate 2-colored graph, then it must be G A and G B -colorable. By Lemma 4, it must be G A × G B -colorable. Note that the product of this two graphs is T -colorable. Thus H is T -colorable. By Lemma 8, the result follows.

Non-degenerate 2-colored bipartite graphs
In this part, we will further classify the non-degenerate 2-colored bipartite graphs. By Lemma 5, the largest possible Turán density of a 2-colored bipartite graph H is 3 2 , so if π(H) < 3 2 , it must be contained in the construction G c and its variations G D , G E , thus it must be colored by the product of these constructions. While the product of graphs generated by the three constructions is a blow-up of following graph H 8 . Let ACX stand for the vertex in A × C × X, similar for other labels: To compute the Turán density of H 8 , we need the following 2-colored graph T 2 = ([4], {12, 14, 23, 24, 34}, {12, 13, 14, 23, 34}). T 2 is not contained in a variation of G C , thus π(T 2 ) ≥ 3 2 . .Thus π({T 1 , T 2 }) ≤ 4 3 .
Proof. It is not hard to check the cases for n ≤ 3. Let n ≥ 4, by induction on n we assume the statement holds for any {T 1 , T 2 }-free graph on less than n vertices. Note if G contains no double-colored edge, the result is trivial. Thus we assume G contains at least one double-colored edge. Then G is one of the following cases.
Case 1: G contains a triangle consisting of three double-colored edges, let V 1 = {a, b, c} be the vertices of this triangle and V 2 = V (G) \ V 1 . By Lemma 7 "Case 2", for any vertex w ∈ V 2 , there are at most 4 edges form w to V 1 .
For any vertex w ∈ V 2 , there are at most 4 edges to V 1 . One can check the following graphs with 5 edges from w to V 1 contain T 1 , T 2 and T 1 respectively. Applying the inductive hypothesis to G[V 2 ], we have Then the number of edges in G is: for the first three cases, for Case 4, The induction step is finished. It follows that π({T 1 , T 2 }) ≤ 4 3 . Lemma 10. π(H 8 ) = 4 3 . Proof. We first prove π(H 8 ) ≤ 4 3 . To show this, we prove that H 8 is T 1 and T 2 -colorable, i.e there are graph homomorphisms from H 8 to T 1 and from H 8 to T 2 . For T 2 : We define a map g by g(ACX) = g(ADX) = 1, g(ADY ) = g(ACY ) = 3, g(BDX) = g(BDY ) = 2, g(BCY ) = g(BCX) = 4. It is easy to check that g is a graph homomorphism from H 8 to T 2 .
For any positive integer n, let G n be a 2-colored graph on n vertices such that h n (G n ) ≥ π(T 1 , T 2 ) + ǫ = π(T 1 (s), T 2 (s)) + ǫ, for any s ≥ 2 and ǫ > 0. Then G n contains T 1 (s) or T 2 (s) as subgraph, further G n contains H 8 as subgraph. Then  Given a family of {2, 3}-graphs H, the Turán density of H is defined to be: where the maximum is taken over all H-free hypergraphs G on n vertices satisfying G ⊆ K   In [4], we say a degenerate R-graph is trivial if it is a sub-graph of a blow-up of the chain C R . By Theorem 4, there exist non-trivial degenerate {2, 3}-graphs. The {2, 3}-graph H = {12, 123} is a chain, thus it is degenerate. By Theorem 3, the subdivision H ′ = {14, 24, 123} is also degenerate, but it is non-trivial. As showed in [11], H 0 = S(K 1,2 2 ) = {13, 12, 123} is not degenerate, and π(H 0 ) = 5 4 . So what does the degenerate {2, 3}-graph look like? To answer this question, we may need to construct a family of {2, 3}-graphs G n with h n (G n ) > (1 + ǫ) for some ǫ > 0. Here are three {2, 3}-graphs with edge density greater than 1.
Note that for any R-graph H (with possible loops), one can construct the family of H-colorable R-graph by blowing up H in certain way. The langrangian of H is the maximum edge density of the H-colorable R-graph that one can get this way. For more details of R-graphs with loops, blow-up, and Lagrangian, please refer to [4]. In this part, we will use an easy-understood way to calculate the edge densities.
Let |A| = xn and |B| = |C| = 1−x 2 n for some value x ∈ (0, 1). We have h n (G Let |X| = xn and |Y | = (1 − x)n for some value x ∈ (0, 1), we have The following lemma shows a relation between such {2, 3}-graphs and the 2-colored graphs and can help us determine the upper bound for the Turán density of some {2, 3}-graphs. Proof. Let n be positive integer, let G = (V, E 2 (G), E 3 (G)) be an arbitrary H ′ -free {2, 3}-graph on n vertices. For any vertex v ∈ V (G), let G v = (V (G) \ {v}, E v,2 , E v,3 ) be a 2-colored graph obtained form G, such that the red edges are E v,2 = E 2 (G), the blue edges are E v,3 = {u, w|{vuw} ∈ E 3 }. Observe that G v is H-free since G is H ′ -free. Thus h n−1 (G v ) ≤ π n (H). Since