Monochromatic k -connected Subgraphs in 2-edge-colored Complete Graphs
aa r X i v : . [ m a t h . C O ] S e p Monochromatic k -connected Subgraphs in 2-edge-coloredComplete Graphs Chunlok Lo a , Hehui Wu b,1 , Qiqin Xie c a University of Alberta, Edmonton, AB, Canada T6G 2R3 b Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China 200438 c Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China 200438
Abstract
Bollob´as and Gy´arf´as conjectured that for any k, n ∈ Z + with n > k − n vertices leads to a k -connected monochromatic subgraphwith at least n − k +2 vertices. We find a counterexample with n = 5 k − ⌈√ k − ⌉−
3, thusdisproving the conjecture, and we show the conjecture is true for n ≥ k − min {√ k − , . k + 4 } . Keywords:
Connectivity, Monochromatic
1. Introduction
Ramsey theory is one of the most important research areas in combinatorics. For anygiven integers s, t , the Ramsey number R ( s, t ) is the smallest integer n , such that for any2-edge-colored (red/blue) K n , there must exist a red K s or a blue K t . In 1930, Ramsey [17]proved the existence of Ramsey numbers. However, estimating Ramsey numbers is knownto be notoriously challenging.There are many variations of the original Ramsey problem, including the one consideringhighly-connected subgraphs instead of cliques. A graph is k -connected if and only if it hasmore than k vertices and does not have a vertex cut of size at most k −
1. Let r c ( k ) denotethe smallest integer such that every c -edge-colored complete graph on r c ( k ) vertices mustcontain a k -connected monochromatic subgraph. In 1983, Matula [16] proved 2 c ( k − ≤ r c ( k ) < (10 / c ( k −
1) + 1. Moreover, for 2-edge-coloring, Matula [16] improved the upperbound to r ( k ) < (3 + p / k −
1) + 1. However, Matula’s result does not have anyrestriction on the order of the k -connected monochromatic subgraph. Bollob´as and Gy´arf´as[1] proposed the following conjecture: Conjecture 1.1.
Let k, n be positive integers. For n > k − , every 2-edge-colored K n contains a k -connected monochromatic subgraph with at least n − k + 2 vertices. Note that the statement is not true for n ≤ k −
1) by Matula’s result [16]. (Alsosee [1].) Moreover, no matter how large n is, n − k + 2 is the best possible bound forthe order of the k -connected subgraph by the example B ( n, k ) in [1]. Besides proposing Email addresses: [email protected] (Chunlok Lo), [email protected] (Hehui Wu), [email protected] (Qiqin Xie) Partially supported by NSFC 11931006 and the Dawn Program of Shanghai Education Commission.
Preprint submitted to Elsevier September 8, 2020 he conjecture, Bollob´as and Gy´arf´as verified the conjecture for k ≤
2, and showed it issufficient to prove the conjecture holds for 4 k − ≤ n < k −
5. Liu, Morris, and Prince[13] verified the conjecture for k = 3, and proved it for n ≥ k −
15. Later, Fujita andMagnant [6] improved the bound to n > . k − K n with no large k -connected monochromatic subgraphs has also been studied.(See [8].)The main result of this paper is that we show Conjecture 1.1 fails for n = 5 k − ⌈√ k − ⌉ −
3. On the other hand, we verify the conjecture for larger n . Theorem 1.2. For every k ∈ Z + , let n = 5 k − ⌈√ k − ⌉ − . There exists a 2-edge-colored K n ,such that there is no k -connected monochromatic subgraph, which contains at least n − k + 2 vertices. Let k, n ∈ Z + . If n ≥ k − √ k − − and n ≥ k − . k − , then for any 2-edge-colored K n , there exists a k -connected monochromatic subgraph, which containsat least n − k + 2 vertices. Note that when k = 3 ,
5, or 7, we will always have ⌊√ k − ⌋ + 3 ≤ ⌊ . k ⌋ + 4. Besides,since ⌊√ k − ⌋ + 3 ≥ ⌊ . k ⌋ + 4 ≥ k ∈ Z + , the statement always holds for n ≥ k − k -connected graph has minimum degree at least k , Theorem 1.2 (2) leads to thefollowing corollary: Corollary 1.3. If n ≥ k − √ k − − and n ≥ k − . k − , then for any 2-edge-colored K n , there exists a monochromatic subgraph with minimum degree at least k , which containsat least n − k + 2 vertices. This problem concerning monochromatic large subgraphs with a specified minimum de-gree in edge-colored graphs has been studied by Caro and Yuster [2]. By applying theirconclusion on 2-edge-colored complete graphs, the corollary holds when n ≥ k + 4, whichcould be covered by our result.In Section 2, we will give some definitions and lemmas related to connectivity. In Section3, we will prove Theorem 1.2 (1), and in Section 4, we will prove Theorem 1.2 (2). The restof this section will be devoted to terminologies and notations. We follow the notations andterminologies for graphs from [3].Given a graph G and an edge-coloring of G with colors red and blue, let R be thespanning subgraph induced by red edges, and B be the spanning subgraph induced by blueedges.We use δ ( G ) to denote the minimum degree of G . Given vertex sets V , V ⊆ V ( G ), let e G ( V , V ) be the number of edges with one endpoint in V and the other endpoint in V .For S ⊆ V ( G ), we use N G ( S ) to denote the vertex set { v : v / ∈ S, ∃ u ∈ S, uv ∈ E ( G ) } , and2 G [ S ] to denote S ∪ N G ( S ). We use G ( S ) to denote the subgraph of G induced by S , and G − S to denote the subgraph of G induced by V ( G ) \ S . Let e = uv where u, v ∈ V ( G )and e / ∈ E ( G ). We use G + e to denote the graph ( V ( G ) , E ( G ) ∪ { e } ).For S ⊆ V ( G ), we say S is complete in G if G ( S ) is a complete subgraph of G . Fordisjoint V , V ⊆ V ( G ), we say [ V , V ] is complete in G if G has a complete bipartitesubgraph with partite sets V , V .
2. Graphs without large k -connected subgraphs In this section, we will introduce a decomposition for graphs with no k -connected sub-graphs of large order. Definition 2.1.
Let k ∈ Z + , f ( k ) be a non-negative function on k . Let G be a graph on n vertices, where n ≥ f ( k ) + k . We define an ( f ( k ) , k ) -decomposition of G to be a sequenceof triples { ( A i , C i , D i ) } , i ∈ [1 , l ], such that1. V ( G ) is a disjoint union of A , C , D C i ∪ D i is a disjoint union of A i +1 , C i +1 , D i +1 , i ∈ [1 , l − | C i | ≤ k −
1, 1 ∈ [1 , l ]4. 1 ≤ | A i | ≤ | D i | , and there is no edge between A i and D i , i ∈ [1 , l ]5. | C i | + | D i | ≥ n − f ( k ), i ∈ [1 , l − | C l | + | D l | < n − f ( k )By (1) and (2) of Definition 2.1, we have: Proposition 2.2. V ( G ) is a disjoint union of A , . . . , A i , C i , D i for any i ∈ [1 , l ] . We can also find a partition for the edges in G with respect to the decomposition: Proposition 2.3. E ( G ) is a disjoint union of E AA , E AC , E l , where E AA contains alledges with both endpoints in A i for some i ∈ [1 , l ] , E AC contains all edges with one endpointin A i , and the other in C i for some i ∈ [1 , l ] , and E l contains all edges with both endpointsin C l ∪ D l .Proof. Let e = uv ∈ E ( G ). Suppose there exists i ∈ [1 , l ], such that { u, v } ∩ A i = ∅ , wetake the smallest such i . By symmetry, we may assume u ∈ A i , then by (4) of Definition2.1, either v ∈ A i or v ∈ C i . Thus e ∈ E AA or e ∈ E AC . Suppose there does not exist such i , then by Proposition 2.2, u, v ∈ C l ∪ D l , hence e ∈ E l . Lemma 2.4.
Let k ∈ Z + , f ( k ) be a non-negative function on k . Let G be a graph on n vertices with n ≥ f ( k ) + k . If G does not have a k -connected subgraph with at least n − f ( k ) vertices, then G has an ( f ( k ) , k ) -decomposition.Proof. Let G = G . Since f ( k ) is non-negative, | G | = n ≥ n − f ( k ). We repeat thefollowing steps until | G i | < n − f ( k ).1. Let C i +1 be a cut of G i of size at most k −
1. Since | G i | ≥ n − f ( k ) ≥ k , there mustexist one such cut.2. Let A i +1 be the vertex set of smallest component of G i − C i +1 , and D i +1 = V ( G i ) \ ( A i +1 ∪ C i +1 ).3. Let G i +1 be the subgraph of G i induced by C i +1 ∪ D i +1 .3he sequence of triples generated by the above procedure is an ( f ( k ) , k )-decomposition of G . Definition 2.5.
We say an ( f ( k ) , k )-decomposition is strong if | A i | + | C i | < n − f ( k ), forany i ∈ [1 , l ]. Lemma 2.6.
Let k ∈ Z + , f ( k ) be a non-negative function on k . Let G be a graph on n vertices, where n ≥ f ( k ) + k . If G has a strong ( f ( k ) , k ) -decomposition, then G does nothave a k -connected subgraph with at least n − f ( k ) vertices.Proof. Let { ( A i , C i , D i ) } , i ∈ [1 , l ] be a strong ( f ( k ) , k )-decomposition of G . Suppose G has a k -connected subgraph H such that | H | ≥ n − f ( k ). Let i ∗ be the smallest i such that A i ∩ V ( H ) = ∅ . Note that by Proposition 2.2 and (6) of Definition 2.1, such i ∗ must exist.Then H must be a subgraph of G ( A i ∗ ∪ C i ∗ ∪ D i ∗ ). We claim V ( H ) ∩ D i ∗ = ∅ . Otherwiseby (3) and (4) of Definition 2.1, V ( H ) ∩ C i ∗ is a cut of H of size at most k −
1, which isa contradiction to the connectivity of H . Thus H must be a subgraph of G ( A i ∗ ∪ C i ∗ ).However since the decomposition is strong, | H | ≤ | A i ∗ | + | C i ∗ | < n − f ( k ). We concludethat G does not have a k -connected subgraph with at least n − f ( k ) vertices.
3. The counterexample
In this section, we will demonstrate and verify the counterexample in Theorem 1.2 (1).
Proof of Theorem 1.2 (1).
Let τ = ⌈√ k − ⌉ , then n = 5 k − τ −
3. We may assume n ≥ k −
3, otherwise Theorem 1.2 (1) holds by the example B ( k ) given in [1]. (Also see[16].) Thus we assume τ ∈ (0 , . k ], as 5 k − τ − ≥ k − V ( G ) = A ∪ A ∪ · · · ∪ A k +1 ∪ C k +1 ∪ D k +1 , where A i = { a i } for i ∈ [1 , k ], A k +1 = { a l , . . . , a k − l } , C k +1 = { c l , . . . , c k − l } , and D k +1 = { d l , . . . , d k − τ − l } . Thus n = | V ( G ) | = P k +1 i =1 | A i | + | C k +1 | + | D k +1 | = k +( k − k − k − τ −
1) = 5 k − τ − C i and D i for i ∈ [1 , k ]. Let ALU = { a l , . . . , a τl } , and DLU = { d l , . . . , d k − τl } . Let S = { s i } be the sequence 1 , . . . , τ, , . . . , τ, . . . , , . . . , τ (repeat τ times).Since τ = ⌈√ k − ⌉ , the length of S is at least τ = ⌈√ k − ⌉ ≥ k −
1. Note that thereis no i ∈ [1 , k −
1] such that s i − = s i . We set C i = ( ALU \ { a s i − l , a s i l } ) ∪ { c il } ∪ DLU for i ∈ [1 , k − C k = ( ALU \ { a s k − l } ) ∪ DLU , D = V ( G ) \ ( A ∪ C ), and D i = C i − ∪ D i − \ ( A i ∪ C i ) for i ∈ [2 , k ].We color all edges between A i and D i blue for all i ∈ [1 , k + 1], and all the other edgesred. We use R (resp. B ) to denote the spanning subgraph induced by red (resp. blue)edges. Claim 3.1. R does not have a k -connected subgraph with at least n − k + 2 vertices.Proof. We prove this by verifying { ( A i , C i , D i ) } , i ∈ [1 , l ] is a strong (2 k − , k )-decompositionof R . Note that in our example, l = k + 1.By the construction, (1)(2)(5)(6) of Definition 2.1 hold. We will verify 2.1 (3) and (4)for our example.(3) | C i | ≤ k − i ∈ [1 , l ]For i ∈ [1 , k − | C i | = | ALU | − | DLU | = τ − k − τ ) = k − | C k | = | ALU | − | DLU | = τ − k − τ ) = k − | C k +1 | = |{ c l , . . . , c k − l }| = k − ≤ | A i | ≤ | D i | , and there is no edge between A i and D i , i ∈ [1 , l ]4or i ∈ [1 , k ], 1 = | A i | ≤ | D i | . 1 ≤ | A k +1 | = ( k − ≤ k − τ − | D k +1 | as τ ∈ (0 , . k ]. All edges between A i and D i are blue. Thus R does not contain any edgebetween A i and D i for all i ∈ [1 , l ].Moreover, since n = 5 k − τ − ≥ k − | A i | + | C i | = 1 + ( k −
1) = k < n − k + 2for i ∈ [1 , k ], and | A k +1 | + | C k +1 | = ( k −
1) + ( k −
1) = 2 k − < n − k + 2. Thus | A i | + | C i | < n − (2 k −
2) for any i ∈ [1 , l ]. The decomposition is strong.Thus by Lemma 2.6, R does not have a k -connected subgraph with at least n − k + 2vertices. Claim 3.2. B does not have a k -connected subgraph with at least n − k + 2 vertices.Proof. We prove the claim by finding a sequence of vertices u , u , . . . , u k − such that e B ( { u x } , V ( B ) \ { u , . . . , u x } ) ≤ k −
1. That is, there is no subgraph H of B with δ ( H ) ≥ k and order at least n − k + 2.We claim that { u x } = c l , . . . , c k − l , d l , . . . , d k − τl , a l , . . . , a τl is a sequence that satisfiesthe requirement. Note that the sequence contains ( k −
1) + ( k − τ ) + τ = 2 k − ≤ x ≤ k − k − C k +1 has no neighbor in A k +1 ∪ C k +1 ∪ D k +1 and c il ∈ C i , we have e B ( { u x } , V ( B ) \ { u , . . . , u x } ) = | N B ( c xl ) | = |{ a , . . . , a k } \ { a x }| = k − . k ≤ x ≤ k − k − τ ).The next k − τ vertices are all in the set DLU . They are not adjacent to any of the a , . . . , a k , since DLU ⊆ C i for i ∈ [1 , k ]. Moreover, all edges with both endpoints in D k +1 are red. Thus, take x ′ = x − ( k − e B ( { u x } , V ( B ) \ { u , . . . , u x } ) = | N B − C k +1 ( d x ′ l ) | = | A k +1 | = k − . k + ( k − τ ) ≤ x ≤ k − τ vertices are all in ALU . According to the definition of C i for i ∈ [1 , k ],every vertex in ALU is adjacent to at most τ of a , . . . , a k . Indeed, let a jl ∈ ALU ,where j ∈ [1 , τ ]. Then a jl is adjacent to a i in B if and only if j ∈ { s i − , s i } for i ∈ [1 , k − a jl is adjacent to a k in B if and only if j = s k − . Since j occurs τ times in S , at most τ of a , . . . , a k are adjacent to a jl . Moreover, all edges with bothendpoints in A k +1 are red. Thus, take x ′′ = x − ( k − − ( k − τ ), we have e B ( { u x } , V ( B ) \ { u , . . . , u x } ) = | N B − ( C k +1 ∪ DLU ) ( a x ′′ l ) |≤| D k +1 \ DLU | + τ = (2 k − τ − − ( k − τ ) + τ = k − . Thus we conclude that B does not have a subgraph H with at least n − k +2 vertices with δ ( H ) ≥ k . Hence, B does not have a k -connected subgraph of order at least n − k + 2.By Claim 3.1 and 3.2, the 2-coloring we proposed contains no k -connected monochro-matic subgraph of order at least n − k +2, which completes the proof of Theorem 1.2(1).5 . Proof of Theorem 1.2(2) For k, n ∈ Z + , let λ = min {⌊√ k − ⌋ + 3 , ⌊ . k ⌋ + 4 } . We will prove the statementfor n ≥ k − λ . Since 4 k − k ∈ Z + , we will always have λ < √ k − k − λ + 6 λ − >
0, as λ > −√ k − k ∈ Z + . Moreover, we will always have n ≥ k −
3, since ⌊ . k ⌋ + 4 ≤ k + 3 for all k ∈ Z + .Suppose there exists a 2-edge-colored K n , such that there is no k -connected monochro-matic subgraph with at least n − k + 2 vertices. Let R be the red graph, and B be theblue graph. We may assume E ( R ) is maximized.Since R does not have a k -connected subgraph with at least n − k + 2 vertices, byLemma 2.4, R must have a (2 k − , k )-decomposition { ( A i , C i , D i ) } , i ∈ [1 , l ]. By Definition2.1 (4), for any i ∈ [1 , l ], there is no edge between A i and D i in R . Thus [ A i , D i ] is completein B for any i ∈ [1 , l ]. Proposition 4.1.
Suppose exists i ∈ [1 , l ] such that B ( C i ∪ D i ) has a k -connected subgraph H of order at least k − , then B ( A i ∪ V ( H )) is k -connected.Proof. Since H is a subgraph of B ( C i ∪ D i ) and | C i | ≤ k −
1, we have | V ( H ) ∩ D i | = | V ( H ) | − | V ( H ) ∩ C i | ≥ (2 k − − ( k −
1) = k . Since [ A i , V ( H ) ∩ D i ] is complete in B ( A i ∪ V ( H )), B ( A i ∪ V ( H )) is k -connected.By Definition 2.1 (2), C i ∪ D i = A i +1 ∪ C i +1 ∪ D i +1 , i ∈ [1 , l − Corollary 4.2.
Suppose exists i ∈ [1 , l ] such that B ( C i ∪ D i ) has a k -connected subgraph H of order at least k − , then B ( A ∪ A ∪ · · · ∪ A i ∪ V ( H )) is k -connected. Claim 4.3. | A i | ≤ k − , ∀ i ∈ [1 , l ] .Proof. Suppose exists i such that | A i | ≥ k . By (4) of Definition 2.1, | D i | ≥ | A i | ≥ k . Since[ A i , D i ] is complete in B , we have B ( A i ∪ D i ) is k -connected. If i = 1, B ( A ∪ D ) is a k -connected subgraph of B . If i ≥
2, By (2) of Definition 2.1, B ( A i ∪ D i ) is a k -connectedsubgraph of B ( C i − ∪ D i − ). Moreover, | B ( A i ∪ D i ) | = | A i | + | D i | ≥ k + k > k −
1. Thus byapplying Corollary 4.2 on ( i −
1) and H = B ( A i ∪ D i ), we have B ( A ∪ A ∪ · · · ∪ A i ∪ D i ) is k -connected. However, by Proposition 2.2 and (3) of Definition 2.1, | A ∪ A ∪· · ·∪ A i ∪ D i | = | V ( G ) | − | C i | ≥ n − ( k − ≥ n − k + 2, a contradiction.Combining (5)(6) of Definition 2.1 and Proposition 2.2, we have the following corollary: Corollary 4.4. k − ≤ P li =1 | A i | ≤ k − .Proof. By Definition 2.1 (6) and Proposition 2.2, P li =1 | A i | = n − ( | C l | + | D l ) ≥ n − ( n − k +1) = 2 k −
1. By Definition 2.1 (5) and Proposition 2.2, P li =1 | A i | = ( P l − i =1 | A i | )+ | A l | = n − ( | C l − | + | D l − | ) + | A l | ≤ n − ( n − k + 2) + ( k −
1) = 3 k − E ( R ), we have: Claim 4.5. A i is complete in R for any i ∈ [1 , l ] .Proof. Suppose there exists i ∗ ∈ [1 , l ], and u, v ∈ A i ∗ such that uv / ∈ E ( R ). Consider R ′ = R + uv . Since there does not exist i ′ ∈ [1 , l ] such that uv is an edge between A i ′ and D i ′ , { ( A i , C i , D i ) } , i ∈ [1 , l ] is also a (2 k − , k )-decomposition of R ′ . Moreover, by Claim6.3 and (3) of Definition 2.1, for any i ∈ [1 , l ], | A i | + | C i | ≤ ( k −
1) + ( k − ≤ n − k + 2since n ≥ k −
3. Hence, the decomposition is strong, and by Lemma 2.6, R ′ does not havea k -connected subgraph with at least n − k + 2 vertices, a contradiction to the maximalityof E ( R ). Thus A i is complete in R for any i ∈ [1 , l ].Similarly we can prove: Claim 4.6. D l is complete in R . Claim 4.7. C l is complete in R . Claim 4.8. [ A i , C i ] is complete in R for any i ∈ [1 , l ] . Claim 4.9. [ D l , C l ] is complete in R . According to the above 5 claims, we conclude that
Claim 4.10. E ( B ) is the union of all edges between A i and D i for all i ∈ [1 , l ] , and allother edges are in E ( R ) . Since B does not have a k -connected subgraph with at least n − k +2 vertices, by Lemma2.4, B must have a (2 k − , k )-decomposition { ( U x , S x , T x ) } , x ∈ [1 , l B ]. For convenience,we use U to denote U ∪ U ∪ · · · ∪ U l B , and U to denote V ( G ) \ U . Note that by Definition2.1 (3), | S x | ≤ k −
1. Moreover, by a similar proof of Claim 4.3, we will have
Claim 4.11. | U x | ≤ k − . As a corollary, we have
Corollary 4.12. k − ≤ | U | ≤ k − . We will complete the proof by counting the total number of edges between U x and S x .For every i ∈ [1 , l ], we use X i to denote the set of integers x ∈ [1 , l B ] such that A i ∩ U x = ∅ and D i ∩ U x = ∅ , and e U i to denote ∪ x ∈ X i U x . We use E UB (resp. E UR ) to denote the set ofall blue (resp. red) edges between U x and S x for all x ∈ [1 , l B ]. Thus | E UB | + | E UR | ≤ l B X x =1 | U x || S x | ≤ ( k − l B X x =1 | U x | = ( k − | U | . Next, we will show that | E UB | + | E UR | > ( k − | U | .The blue graph, by Claim 4.10, contains all edges between A i and D i for i ∈ [1 , l ]. Onthe other hand, by Proposition 2.3, we can classify the blue edges into 3 types: edges withboth endpoints in U (the E l type), edges with both endpoints in U x for some x ∈ [1 , l B ](the E AA type), and those in E UB , which has one endpoint in U x and the other in S x forsome x ∈ [1 , l B ] (the E AC type). Thus, | E UB | = l X i =1 l B X x =1 ( | A i ∩ U x || D i ∩ S x | + | A i ∩ S x || D i ∩ U x | )= l X i =1 ( | A i || D i | − | A i ∩ U || D i ∩ U | − X x ∈ X i | A i ∩ U x || D i ∩ U x | ) .
7e will first estimate the lower bound for P li =1 ( | A i || D i | − | A i ∩ U || D i ∩ U | ). l X i =1 ( | A i || D i | − | A i ∩ U || D i ∩ U | ) (cid:13) ≥ l X i =1 | A i | ( n − | C i | − i X j =1 | A j | ) − l X i =1 | A i ∩ U | ( n − | U | − i X j =1 | A j ∩ U | ) (cid:13) ≥ (4 k − λ + 1)( l X i =1 | A i | ) − (5 k − λ − | U | )( l X i =1 | A i ∩ U | ) − l X i =1 i X j =1 | A i || A j | + l X i =1 i X j =1 | A i ∩ U || A j ∩ U | =(4 k − λ + 1)( l X i =1 | A i | ) − (5 k − λ − | U | )( l X i =1 | A i ∩ U | ) − ( P li =1 | A i | ) + P li =1 | A i | P li =1 | A i ∩ U | ) + P li =1 | A i ∩ U | k − λ + 1)( l X i =1 | A i | ) − (5 k − λ − | U | )( l X i =1 | A i ∩ U | ) −
12 ( l X i =1 | A i | ) + 12 ( l X i =1 | A i ∩ U | ) − l X i =1 ( | A i | + | A i ∩ U | )( | A i | − | A i ∩ U | )=(3 k − λ + 2)( l X i =1 | A i | ) − (4 k − λ + 1 − | U | )( l X i =1 | A i ∩ U | )+ ( k − l X i =1 | A i | − l X i =1 | A i ∩ U | ) −
12 ( l X i =1 | A i | ) + 12 ( l X i =1 | A i ∩ U | ) − l X i =1 (( | A i ∩ U | + | A i ∩ U | ) + | A i ∩ U | )( | A i ∩ U | )= −
12 ((3 k − λ + 2) − l X i =1 | A i | ) + 12 (3 k − λ + 2) + 12 ((4 k − λ + 1 − | U | ) − l X i =1 | A i ∩ U | ) −
12 (4 k − λ + 1 − | U | ) + ( k − l X i =1 | A i ∩ U | ) − l X i =1 ( | A i ∩ U | + 2 | A i ∩ U | )( | A i ∩ U | ) (cid:13) (cid:13) (cid:13) ≥ −
12 ((3 k − λ + 2) − (2 k − + 12 (3 k − λ + 2) −
12 (3 k − λ + 2 − | U | ) − ( k − k − λ + 2 − | U | ) −
12 ( k − l X i =1 (( k − − | A i ∩ U | )( | A i ∩ U | ) − l X i =1 | A i ∩ U | (cid:13) (cid:13) ≥ −
12 ((3 k − λ + 2) − (2 k − + 12 (3 k − λ + 2) −
12 ((3 k − λ + 2) − (2 k − − ( k − k − λ + 2) −
12 ( k − + ( k − | U | + l X i =1 (( k − − | A i ∩ U | )( | A i ∩ U | ) − l X i =1 | A i ∩ U | = − ( k − λ + 3) + 12 (3 k − λ + 2) − ( k − k − λ + 2) −
12 ( k − + ( k − | U | + l X i =1 (( k − − | A i ∩ U | )( | A i ∩ U | ) − l X i =1 | A i ∩ U | = 12 (4 k − λ + 6 λ −
11) + ( k − | U | + l X i =1 (( k − − | A i ∩ U | )( | A i ∩ U | ) − l X i =1 | A i ∩ U | (cid:13) > ( k − | U | + l X i =1 (( k − − | A i ∩ U | )( | A i ∩ U | ) − l X i =1 | A i ∩ U | Notes: (cid:13) By Proposition 2.2, and | C i ∩ U | is omitted. (cid:13) n ≥ k − λ , | A i | ≥ | A i ∩ U | , and | C i | ≤ k − (cid:13) λ ≤ . k + 4. (cid:13) k − ≤ P li =1 | A i | ≤ k − (cid:13) (1 / k − λ + 1 − | U | ) − P li =1 | A i ∩ U | ) is omitted. (cid:13) k − ≤ | U | ≤ k − (cid:13) k − λ + 6 λ − > Next, we will find an upper bound for the summation of | A i ∩ U x || D i ∩ U x | . We remarkthat for every i ∈ [1 , l ], X i is the set of integers x ∈ [1 , l B ] such that A i ∩ U x = ∅ and D i ∩ U x = ∅ . Moreover, e U i = ∪ x ∈ X i U x . By Claim 4.11, | D i ∩ U x | ≤ | U x | − | A i ∩ U x | ≤ ( k − − | A i ∩ U x | . Thus, l X i =1 X x ∈ X i | A i ∩ U x || D i ∩ U x | ≤ l X i =1 ( k − | A i ∩ e U i | − l X i =1 X x ∈ X i | A i ∩ U x | In the red graph, for i ∈ [1 , l ], suppose there exists x ∈ X i . By the choice of X i , D i ∩ U x = ∅ . Since [ A i , D i ] is complete in blue, we must have A i ∩ U ⊆ S x , and A i ∩ U x ′ ⊆ S x for all x ′ ∈ X i , x ′ > x . Since A i is complete in red, all edges between A i ∩ U x and A i ∩ S x are red. Thus | E UR | ≥ l X i =1 | A i ∩ U || A i ∩ e U i | + l X i =1 X x ,x ∈ X i x
12 ( l X i =1 | A i ∩ U | − l X i =1 | A i ∩ e U i | ) + 12 l X i =1 X x ∈ X i | A i ∩ U x | (cid:13) ≥ ( k − | U | + l X i =1 (( k − − | A i ∩ U | )( | A i ∩ U | − | A i ∩ e U i | ) − l X i =1 ( | A i ∩ U | + | A i ∩ e U i | )( | A i ∩ U | − | A i ∩ e U i | ) ≥ ( k − | U | + l X i =1 (( k − − | A i ∩ U | )( | A i ∩ U | − | A i ∩ e U i | ) − l X i =1 ( | A i ∩ U | + | A i ∩ U | )( | A i ∩ U | − | A i ∩ e U i | )=( k − | U | + l X i =1 (( k − − | A i ∩ U | − | A i ∩ U | )( | A i ∩ U | − | A i ∩ e U i | ) (cid:13) ≥ ( k − | U | Notes: (cid:13) (1 / P li =1 P x ∈ X i | A i ∩ U x | is omitted. (cid:13) | A i ∩ U | + | A i ∩ U | = | A i | ≤ k − which is a contradiction to | E UB | + | E UR | ≤ ( k − | U | . Thus, G must have a k -connectedmonochromatic subgraph with at least n − k + 2 vertices.
5. Conclusion
In this paper, we presented a counterexample of Bollob´as and Gy´arf´as’ conjecture with n = 5 k − ⌈√ k − ⌉ −
3. We also verified the conjecture for n ≥ k − λ , where λ =10in {⌊√ k − ⌋ + 3 , ⌊ . k ⌋ + 4 } . Remark λ = ⌊√ k − ⌋ + 3 when k = 3 ,
5, or 7. However,there is still a Θ( √ k ) gap between the two bounds. We have found some examples, whichare very different from the counterexample we raised in section 3, and are not covered bythe inequality in section 4. However, none of them could serve as a counterexample to thestatement. Thus, we conjecture that: Conjecture 5.1.
Let k, n ∈ Z + . If n > k − ⌊ √ k − ⌋ − , then for any 2-edge-colored K n , there exists a k -connected monochromatic subgraph, which contains at least n − k + 2 vertices. More generally, consider the statement “ For k, n ∈ Z + with n ≥ g ( k ), every 2-edge-colored K n must contain a k -connected monochromatic subgraph with at least n − f ( k )vertices”. For a given f ( k ) ≥ k −
2, what is the minimum g ( k ) for the statement to betrue? Note that when f ( k ) ≤ k −
1, the example B ( n, k ) in [1] can always serve as acounterexample of the statement. On the other hand, if g ( k ) ∈ [4 k − , k −
4] is fixed, whatis the correlated f ( k )? In other words, given the number of vertices n , what is the order ofthe largest k -connected monochromatic subgraph we can guarantee in a 2-edge-colored K n ?Furthermore, there are some open problems related to Bollob´as and Gy´arf´as’ conjecture,such as the multicoloring version of the conjecture, and forcing large highly connected sub-graphs with given independence number. We believe the decomposition we introduced inthis paper could also be applied to improve the results of those topics.
6. Acknowledgments
We are immensely grateful to Henry Liu, Sun Yat-sen University, for his comments onan earlier version of the manuscript. We would like to show our gratitude to Prof. XingxingYu, Georgia Institute of Technology, for sharing his pearls of wisdom with us during thecourse of this research.
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