On Interval Colorings of Complete k-partite Graphs K_{n}^{k}

Mathematical Problems of Computer Science 26, 2006, 28–32. On In terv al Colorings of Complete k − partite Graphs K k n Rafael R. Kamalia n and Petros A. Petrosyan Institute for I nformatics and Automation Problems (I IA P) of NAS of RA e-mails rrk amalian@yahoo.com, p et p etros@ya ho o.com Abstract Problems of existence, c o nstruction and estimation of par ameters of in terv al colorings of complete k − pa rtite graphs K k n are inv estig ated. Let G = ( V , E ) be an undirected graph witho ut lo ops and m ultiple edges [1 ], V ( G ) and E ( G ) b e the sets of vertices and edges of G , resp ectively . The degr ee of a vertex x ∈ V ( G ) is denoted by d G ( x ), the max im um degr ee of a vertex of G -by ∆( G ), a nd the chromatic index [2] of G -by χ ′ ( G ). A g raph is re g ular, if all its vertices hav e the sa me de g ree. If α is a pro per edge coloring of the graph G [3], then the color of an edge e ∈ E ( G ) in the colo r ing α is deno ted by α ( e ), and the set of colors of the edges that are inciden t to a vertex x ∈ V ( G ) , is denoted by S ( x, α ). F or a no n-empt y subset D of Z + , let l ( D ) a nd L ( D ) be the minimal and maximal element o f D , resp ectively . A non-empty subset D of Z + is interv al, if l ( D ) ≤ t ≤ L ( D ) , t ∈ Z + implies that t ∈ D . Interv al D is referred to b e ( q , h )-interv al if l ( D ) = q , | D | = h a nd is denoted by I nt ( q , h ). F or int er v als D 1 and D 2 with | D 1 | = | D 2 | = h , and a p ∈ Z + , the notation D 1 ⊕ p = D 2 means: l ( D 1 ) + p = l ( D 2 ). A prop er coloring α of edg es of G with colo rs 1 , 2 , . . . , t is an interv al t - coloring of G [4 ], if for ea ch colo r i, 1 ≤ i ≤ t, ther e exists at lea st one edge e i ∈ E ( G ) with α ( e i ) = i , a nd the edg es incident with ea ch vertex x ∈ V ( G ) are color ed by d G ( x ) co nsecutive colors . A g raph G is interv al co lorable, if there is t ≥ 1 , for which G has an interv al t -coloring . The set of a ll int er v al colorable gr aphs is denoted by N [5]. F or G ∈ N the lea s t and the gr eatest v alues of t , for which G has an interv al t -coloring , is denoted by w ( G ) a nd W ( G ), r espe c tiv ely . In [5] it is proved: Theorem 1 . Let G b e a reg ular graph. 1) G ∈ N iff χ ′ ( G ) = ∆( G ). 2) If G ∈ N and ∆( G ) ≤ t ≤ W ( G ), then G has an interv al t -coloring . Theorem 2 [6]. Let n = p · 2 q , where p is odd, and q ∈ Z + . Then W ( K 2 n ) ≥ 4 n − 2 − p − q . In this pap er interv al colo r ings of complete k -partite graphs K k n [7] a re in- vestigated, where 1 V  K k n  = n x ( i ) j | 1 ≤ i ≤ k , 1 ≤ j ≤ n o , E  K k n  = n x ( i ) p , x ( j ) q  | 1 ≤ i < j ≤ k , 1 ≤ p ≤ n, 1 ≤ q ≤ n o . It is not har d to s ee that ∆( K k n ) = ( k − 1) · n . F rom the results of [8 ] we imply that χ ′ ( K k n ) =  ( k − 1) · n, if n · k is even, ( k − 1) · n + 1 , if n · k is o dd. Theorem 1 implies: Corollary 1. 1) K k n ∈ N , if n · k is even; 2) K k n / ∈ N , if n · k is o dd. Corollary 2. If n · k is even, then w ( K k n ) = ( k − 1) · n . Theorem 3. If k is even, then W  K k n  ≥  3 2 k − 1  · n − 1. Pro of. Let V  K k n  = n x ( i ) j | 1 ≤ i ≤ k , 1 ≤ j ≤ n o , E  K k n  = n x ( i ) p , x ( j ) q  | 1 ≤ i < j ≤ k , 1 ≤ p ≤ n, 1 ≤ q ≤ n o . F or the gra ph K k n define an edge colo ring λ as follows: for i = 1 , ...,  k 4  , j = 2 , ..., k 2 , i < j, i + j ≤ k 2 + 1 , p = 1 , ..., n, q = 1 , ..., n set: λ  x ( i ) p , x ( j ) q  = ( i + j − 3) · n + p + q − 1; for i = 2 , ..., k 2 − 1 , j =  k 4  + 2 , ..., k 2 , i < j, i + j ≥ k 2 + 2 , p = 1 , ..., n, q = 1 , ..., n set: λ  x ( i ) p , x ( j ) q  =  i + j + k 2 − 4  · n + p + q − 1; for i = 3 , ..., k 2 , j = k 2 + 1 , ..., k − 2 , j − i ≤ k 2 − 2 , p = 1 , ..., n, q = 1 , ..., n set: λ  x ( i ) p , x ( j ) q  =  k 2 + j − i − 1  · n + p + q − 1; for i = 1 , ..., k 2 , j = k 2 + 1 , ..., k , j − i ≥ k 2 , p = 1 , ..., n, q = 1 , ..., n set: λ  x ( i ) p , x ( j ) q  = ( j − i − 1) · n + p + q − 1; 2 for i = 2 , ..., 1 +  k − 2 4  , j = k 2 + 1 , ..., k 2 +  k − 2 4  , j − i = k 2 − 1 , p = 1 , ..., n, q = 1 , ..., n set: λ  x ( i ) p , x ( j ) q  = (2 i − 3) · n + p + q − 1; for i =  k − 2 4  + 2 , ..., k 2 , j = k 2 + 1 +  k − 2 4  , ..., k − 1 , j − i = k 2 − 1 , p = 1 , ..., n, q = 1 , ..., n set: λ  x ( i ) p , x ( j ) q  = ( i + j − 3) · n + p + q − 1; for i = k 2 + 1 , ..., k 2 +  k 4  − 1 , j = k 2 + 2 , ..., k − 2 , i < j, i + j ≤ 3 2 k − 1 , p = 1 , ..., n, q = 1 , ..., n set: λ  x ( i ) p , x ( j ) q  = ( i + j − k − 1) · n + p + q − 1 ; for i = k 2 + 1 , ..., k − 1 , j = k 2 +  k 4  + 1 , ..., k , i < j, i + j ≥ 3 2 k , p = 1 , ..., n, q = 1 , ..., n set: λ  x ( i ) p , x ( j ) q  =  i + j − k 2 − 2  · n + p + q − 1 . Let us show that λ is an interv al  3 2 k − 1  · n − 1  − color ing of the g r aph K k n . First of all let us s how that fo r i = 1 , 2 , ...,  3 2 k − 1  · n − 1 there is an edg e e i ∈ E  K k n  such that λ ( e i ) = i . Consider the vertices x (1) 1 , x (1) 2 , ..., x (1) n , x ( k ) 1 , x ( k ) 2 , ..., x ( k ) n . It is not hard to see that for j = 1 , 2 , ..., n S  x (1) j , λ  = k − 1 [ l =1 ( I nt ( j, n ) ⊕ n · ( l − 1 )) and S  x ( k ) j , λ  = 3 2 k − 2 [ l = k 2 ( I nt ( j, n ) ⊕ n · ( l − 1 )) . Let C and C − b e the s ubsets of colors of the edges , that ar e inciden t to the vertices x (1) 1 , x (1) 2 , ..., x (1) n and x ( k ) 1 , x ( k ) 2 , ..., x ( k ) n in a coloring λ , res pectively , that is: 3 C = n [ j =1 S  x (1) j , λ  and C = n [ j =1 S  x ( k ) j , λ  . It is not hard to see that C ∪ C =  1 , 2 , . . . ,  3 2 k − 1  · n − 1  , and, therefor e, for i = 1 , 2 , ...,  3 2 k − 1  · n − 1 there is an edge e i ∈ E  K k n  such that λ ( e i ) = i . Now, let us s how that the edg es that are incident to a vertex v ∈ V  K k n  are color ed by ( k − 1) · n consecutive colo rs. Let x ( i ) j ∈ V  K k n  , wher e 1 ≤ i ≤ k , 1 ≤ j ≤ n . Case 1. 1 ≤ i ≤ 2 , 1 ≤ j ≤ n . It is not har d to s ee that S  x (1) j , λ  = S  x (2) j , λ  = k − 1 [ l =1 ( I nt ( j, n ) ⊕ n · ( l − 1 )) = I nt ( j, ( k − 1) · n ). Case 2. 3 ≤ i ≤ k 2 , 1 ≤ j ≤ n . It is not har d to s ee that S  x ( i ) j , λ  = k − 3+ i [ l = i − 1 ( I nt ( j, n ) ⊕ n · ( l − 1 )) = I nt ( j + n · ( i − 2 ) , ( k − 1) · n ). Case 3. k 2 + 1 ≤ i ≤ k − 2 , 1 ≤ j ≤ n . It is not har d to s ee that S  x ( i ) j , λ  = k 2 − 1+ i [ l = i − k 2 +1 ( I nt ( j, n ) ⊕ n · ( l − 1 )) = I nt  j + n ·  i − k 2  , ( k − 1) · n  . Case 4. k − 1 ≤ i ≤ k , 1 ≤ j ≤ n . It is not har d to s ee that S  x ( k − 1) j , λ  = S  x ( k ) j , λ  = 3 2 k − 2 [ l = k 2 ( I nt ( j, n ) ⊕ n · ( l − 1 )) = I nt  j + n ·  k 2 − 1  , ( k − 1) · n  . Theorem 3 is pr ov ed. Corollary 3. If k is even and ( k − 1 ) · n ≤ t ≤  3 2 k − 1  · n − 1 , then K k n has an interv al t − coloring. Theorem 4. Let k = p · 2 q , where p is o dd, and q ∈ N . Then W  K k n  ≥ (2 k − p − q ) · n − 1. 4 Pro of. Let V  K k n  = n x ( i ) j | 1 ≤ i ≤ k , 1 ≤ j ≤ n o , E  K k n  = n x ( i ) r , x ( j ) s  | 1 ≤ i < j ≤ k , 1 ≤ r ≤ n, 1 ≤ s ≤ n o . Consider the gra ph K k , where V ( K k ) = { u 1 , u 2 , . . . , u k } , E ( K k ) = { ( u i , u j ) | 1 ≤ i < j ≤ k } . Theorem 2 implies that if k = p · 2 q , where p is o dd, and q ∈ N , then W ( K k ) ≥ 2 k − 1 − p − q . Suppo se ϕ is an int erv al (2 k − 1 − p − q ) − coloring of K k . Define a colo r ing ψ of the edges o f K k n as follows: F or i = 1 , 2 , . . . , k and j = 1 , 2 , . . . , k , i 6 = j set: ψ  x ( i ) r , x ( j ) s  = ( ϕ (( u i , u j )) − 1) · n + r + s − 1 , where r = 1 , 2 , ..., n, s = 1 , 2 , ..., n . Let us show that ψ is an in terv al ((2 k − p − q ) · n − 1 ) − color ing of the g raph K k n . The definition o f ψ and the equalities L ( S ( u i , ϕ )) − l ( S ( u i , ϕ )) = k − 2, i = 1 , 2 , . . . , k imply tha t: 1) S  x ( i ) j , ψ  = L ( S ( u i ,ϕ )) [ m = l ( S ( u i ,ϕ )) ( I nt ( j, n ) ⊕ n · ( m − 1)) = = I nt ( j + n · ( l ( S ( u i , ϕ )) − 1) , ( k − 1) · n ) for i = 1 , 2 , . . . , k and j = 1 , 2 , . . . , n ; 2) k [ i =1 n [ j =1 S  x ( i ) j , ψ  = I nt (1 , (2 k − p − q ) · n − 1). This show that ψ is an interv a l ((2 k − p − q ) · n − 1) − colo ring o f the g raph K k n . Theorem 4 is pr ov ed. Corollary 4. Let k = p · 2 q , w he r e p is o dd a nd q ∈ N . 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