Erdős-Gallai-type results for the rainbow disconnection number of graphs
aa r X i v : . [ m a t h . C O ] J a n Erd˝os-Gallai-type results for the rainbowdisconnection number of graphs ∗ Xuqing Bai , Xueliang Li , Center for Combinatorics and LPMCNankai University, Tianjin 300071, ChinaEmail: [email protected], [email protected] School of Mathematics and Statistics, Qinghai Normal UniversityXining, Qinghai 810008, China
Abstract
Let G be a nontrivial connected and edge-colored graph. An edge-cut R of G is called a rainbow cut if no two edges of it are colored with a same color. Anedge-colored graph G is called rainbow disconnected if for every two distinctvertices u and v of G , there exists a u − v rainbow cut separating them. For aconnected graph G , the rainbow disconnection number of G , denoted by rd ( G ),is defined as the smallest number of colors that are needed in order to make G rainbow disconnected. In this paper, we will study the Erd˝os-Gallai-typeresults for rd ( G ), and completely solve them. Keywords: rainbow cut, rainbow disconnection coloring (number), Erd˝os-Gallai-type result
AMS subject classification 2010:
All graphs considered in this paper are simple, finite and undirected. Let G =( V ( G ) , E ( G )) be a nontrivial connected graph with vertex set V ( G ) and edge set E ( G ). For v ∈ V ( G ), let N G ( v ) and N G [ v ] denote the open neighbour of v and the closed neighbour of v in G , respectively. For any notation or terminology not definedhere, we follow those used in [2]. ∗ Supported by NSFC No.11871034, 11531011 and NSFQH No.2017-ZJ-790. K n to denote a complete graph of order n . A k -factor of G is a k -regular spanning subgraph of G , and G is k -factorable if thereare edge-disjoint k -factors H , H , . . . , H n such that G = H ∪ H ∪ . . . ∪ H n . Asubset M of E is called a matching of G if any two edges of M do not share a commonvertex of G .Let G be a graph with an edge-coloring c : E ( G ) → [ k ] = { , , ..., k } , k ∈ N ,where adjacent edges may be colored the same. When adjacent edges of G receivedifferent colors under c , the edge-coloring c is called proper . The chromatic index of G , denoted by χ ′ ( G ), is the minimum number of colors needed in a proper coloringof G . By a famous theorem of Vizing [14], one has∆( G ) ≤ χ ′ ( G ) ≤ ∆( G ) + 1for every nonempty graph G .A path is called rainbow if no two edges of it are colored the same. An edge-coloredgraph G is called rainbow connected if every two distinct vertices of G are connectedby a rainbow path in G . An edge-coloring under which G is rainbow connected iscalled a rainbow connection coloring . Clearly, if a graph is rainbow connected, itmust be connected. For a connected graph G , the rainbow connection number of G ,denoted by rc ( G ), is the smallest number of colors that are needed to make G rainbowconnected. Rainbow connection was introduced by Chartrand et al. [5] in 2008. Formore details on rainbow connection, see the book [10] and the survey papers [9, 11].In this paper, we investigate a new concept that is somewhat reverse to rainbowconnection. This concept of rainbow disconnection of graphs was introduced byChartrand et al. [4] very recently in 2018.An edge-cut of a connected graph G is a set S of edges such that G − S isdisconnected. The minimum number of edges in an edge-cut is defined as the edge-connectivity λ ( G ) of G . We have the well-known inequality λ ( G ) ≤ δ ( G ). For twovertices u and v , let λ ( u, v ) denote the minimum number of edges in an edge-cut S such that u and v lie in different components of G − S . The so-called upper edge-connectivity λ + ( G ) of G is defined by λ + ( G ) = max { λ ( u, v ) : u, v ∈ V ( G ) } .λ + ( G ) is the maximum local edge-connectivity of G , while λ ( G ) is the minimumglobal edge-connectivity of G .An edge-cut R of an edge-colored connected graph G is called a rainbow cut if notwo edges in R are colored the same. A rainbow cut R is said to separate two vertices u and v if u and v belong to different components of G − R . Such rainbow cut is2alled a u − v rainbow cut. An edge-colored graph G is called rainbow disconnected if for every two distinct vertices u and v of G , there exists a u − v rainbow cut in G . In this case, the edge-coloring c is called a rainbow disconnection coloring of G . Similarly, we define the rainbow disconnection number of a connected graph G ,denoted by rd ( G ), as the smallest number of colors that are needed in order to make G rainbow disconnected. A rainbow disconnection coloring with rd ( G ) colors is calledan rd - coloring of G .The Erd˝os-Gallai-type problem is an interesting problem in extremal graph theory,which was studied in [7, 8, 12] for rainbow connection number rc ( G ); in [6] for properconnection number pc ( G ); in [3] for monochromatic connection number mc ( G ), andmany other parameter of graphs in literature. We will study the Erd˝os-Gallai-typeresults for the rainbow disconnection number rd ( G ) in this paper. For given integers k and n with 1 ≤ k ≤ n −
1, the authors in [4] determined theminimum size of a connected graph G of order n with rd ( G ) = k . So, this brings upthe question of determining the maximum size of a connected graph G of order n with rd ( G ) = k . The authors of [4] conjectured and we determined in [1] the maximumsize of a connected graph G of order n with rd ( G ) = k , for odd integer n . But for eveninteger n , it was left without solution. Now we consider the question of determiningthe maximum size of a connected graph G of even order n with rd ( G ) = k and weget the following result. Theorem 2.1
Let k and n be integers with ≤ k ≤ n − and n be even. Then themaximum size of a connected graph G of order n with rd ( G ) = k is ⌊ ( k +1)( n − ⌋ . Before we give the proof of Theorem 2.1, some useful lemmas are stated as follows.
Lemma 2.2 [4] If G is a nontrivial connected graph, then λ ( G ) ≤ λ + ( G ) ≤ rd ( G ) ≤ χ ′ ( G ) ≤ ∆( G ) + 1 . Lemma 2.3 [4]
Let G be a nontrivial connected graph. Then rd ( G ) = 1 if and onlyif G is a tree. Lemma 2.4 [4]
For each integer n ≥ , rd ( K n ) = n − . emark 1: For any integer n ≥ rd ( K n ) = n − rd ( K ) = 1 and rd ( K ) = 2. Lemma 2.5 [13]
Let G be a graph of order n ( n ≥ k + 2 ≥ . If e ( G ) > k +12 ( n − − σ k ( G ) , where σ k ( G ) = P x ∈ V ( G ) d ( x ) ≤ k ( k − d ( x )) , then λ + ( G ) ≥ k + 1 . Lemma 2.6 If n is even, then there exists a k -regular graph G of order n , where ≤ k ≤ n − , that satisfies both of the following conditions:(i) G is -factorable.(ii) there exists a vertex u of G such that G [ N ( u )] can add ⌊ k ⌋ matching edges.Proof. Let G = K n . Since K n is 1-factorable, K n has 2 n − K n , namely n matching edges. Let e , e ,. . . , e n be n removing matching edges of G and let v i, and v i, be the two end vertices of e i .Let u = v n, be a vertex of G . Then the remaining graph G is a (2 n − N G ( u ) = { v i, , v i, | ≤ i ≤ n − } , and { e i | ≤ i ≤ n − } are matchingedges which can be added to G [ N ( u )] and the number of matching edges is ⌊ n − ⌋ .Second, we remove the 1-factor from G which contains the edge uv n − , , anddenote remaining graph by G . Obviously, G is a (2 n − N G ( u )is N G ( u ) \ v n − , , and { e i | ≤ i ≤ n − } are matching edges which can be added to G [ N ( u )] and the number of matching edges is ⌊ n − ⌋ .Third, we remove the 1-factor from G which contains the edge uv n − , , anddenote remaining graph by G . Obviously, G is a (2 n − N G ( u )is N G ( u ) \ v n − , , and { e i | ≤ i ≤ n − } are matching edges which can be added to G [ N ( u )] and the number of matching edges is ⌊ n − ⌋ .For G j ( j ≥ j is even, then we remove the 1-factor from G j which containsthe edge uv n −⌊ j ⌋ , ; if j is odd, then we remove the 1-factor from G j which containsthe edge uv n −⌊ j ⌋ , . Obviously, the remaining graph G j +1 is a (2 n − j − N G j +1 ( u ) is N G j ( u ) \ v n −⌊ j ⌋ ,i ( i = 1 or 2), and { e i | ≤ i ≤ n − ⌊ j ⌋ − } are matching edges which can be added to G j +1 [ N ( u )] and the number of matchingedges is ⌊ n − j − ⌋ .Repeating the above process, we can get a k -regular graph G n − k of order n whichis 1-factorable. Furthermore, { e i | ≤ i ≤ n − ⌊ n − k ⌋} are matching edges which canbe added to G n − k [ N ( u )] and the number of matching edges is ⌊ k ⌋ . (cid:3) Now we are ready to give a proof to Theorem 2.1.4 roof of Theorem 2.1:
It is easy to see that the graphs G of maximum sizewith order n and rd ( G ) = k is not more than ( k +1)( n − . Otherwise, rd ( G ) ≥ k + 1by Lemmas 2.2 and 2.5. Now we show that the graphs G of maximum size with evenorder n and rd ( G ) = k is ⌊ ( k +1)( n − ⌋ for 1 ≤ k ≤ n −
1. For k = n −
1, let G = K n .Note that | E ( G ) | = ⌊ ( k +1)( n − ⌋ and rd ( G ) = n − k = 1, let G be a tree. Note that | E ( G ) | = ⌊ ( k +1)( n − ⌋ and rd ( G ) = 1 by Lemma 2.3. Now weconstruct a graph G for 2 ≤ k ≤ n − H k − be a ( k − n . For k ≥ H k − can be selected so that it is 1-factorable and thereexists a vertex u of H k − such that ⌊ k − ⌋ matching edges can be added to N H k − ( u )by Lemma 2.6. Let G be a graph by adding ⌊ k − ⌋ matching edges to N H k − ( u ) andadding n − k edges of { uw | w ∈ V ( H k − ) \ N H k − [ u ] } in H k − . Thus, G is a graph oforder n with | E ( G ) | = ( k − n + ⌊ k − ⌋ + n − k = ⌊ ( k +1)( n − ⌋ . Since χ ′ ( H k − ) = k − c of H k − using colors from [ k − c to an edge-coloring c of G by assigning a fresh color k to all newly added edges in H k − . Note that the set E x of edges incident with x in G is a rainbow set for eachvertex x ∈ V ( G ) \ u . Let p and q be two vertices of G . Then at least one of p and q is not u , say p = u . Since E p is a p − q rainbow cut, c is a rainbow disconnectioncoloring of G using at most k colors. Therefore, rd ( G ) ≤ k . On the other hand, E ( G ) = ⌊ ( k +1)( n − ⌋ > k ( n − since n ≥
3, it follows from Lemmas 2.2 and 2.5 that rd ( G ) ≥ k . (cid:3) rd ( G ) Now we consider the following two kinds of Erd˝os-Gallai-type problems for rd ( G ). Problem A . Given two positive integers n and k with 1 ≤ k ≤ n −
1, compute themaximum integer g ( n, k ) such that for any graph G of order n , if | E ( G ) | ≤ g ( n, k ),then rd ( G ) ≤ k . Problem B . Given two positive integers n and k with 1 ≤ k ≤ n −
1, computethe minimum integer f ( n, k ) such that for any graph G of order n , if | E ( G ) | ≥ f ( n, k )then rd ( G ) ≥ k .It is worth mentioning that the two parameters f ( n, k ) and g ( n, k ) are equivalentto another two parameters. Let t ( n, k )= min {| E ( G ) | : | V ( G ) | = n, rd ( G ) ≥ k } and s ( n, k )= max {| E ( G ) | : | V ( G ) | = n, rd ( G ) ≤ k } . It is easy to see that g ( n, k ) = t ( n, k + 1) − f ( n, k ) = s ( n, k −
1) + 1.We first state two lemmas, which will be used to determine the values of f ( n, k )and t ( n, k ). 5 emma 3.1 [4] For integers k and n with ≤ k ≤ n − , the minimum size of aconnected graph of order n with rd ( G ) = k is n + k − . Note that the following result from [1] is also true for n = 1 ,
3. So we can state itas follows, without n ≥ Lemma 3.2 [1]
Let k and n be integers with ≤ k ≤ n − and n be odd. Then themaximum size of a connected graph G of order n with rd ( G ) = k is ( k +1)( n − . Using Lemma 3.1, we first solve Problem A . Theorem 3.3 g ( n, k ) = n + k − for ≤ k ≤ n − .Proof. It follows from Lemma 3.1 that t ( n, k ) = n + k − ≤ k ≤ n −
1. Thus, g ( n, k ) = t ( n, k + 1) − n + k − (cid:3) Now we come to the solution for Problem B , we get the following result. Theorem 3.4 f ( n, k ) = ⌊ k ( n − ⌋ + 1 for ≤ k ≤ n − .Proof. If n is odd, then s ( n, k ) = ( k +1)( n − for 1 ≤ k ≤ n − f ( n, k ) = s ( n, k −
1) + 1 = k ( n − + 1 = ⌊ k ( n − ⌋ + 1 for 1 ≤ k ≤ n − n is even, then s ( n, k ) = ⌊ ( k +1)( n − ⌋ for 1 ≤ k ≤ n − f ( n, k ) = s ( n, k −
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