Estimates for the number of vertices with an interval spectrum in

arXiv:1205.0131v1 [cs.DM] 1 May 2012

Estimates for the number of vertices with an interval spectrum in proper edge colorings of some graphs R.R. Kamaliana∗ a

Institute for Informatics and Automation Problems, National Academy of Sciences of RA, 0014 Yerevan, Republic of Armenia

A proper edge t-coloring of a graph G is a coloring of edges of G with colors 1, 2, ..., t such that each of t colors is used, and adjacent edges are colored differently. The set of colors of edges incident with a vertex x of G is called a spectrum of x. A proper edge t-coloring of a graph G is interval for its vertex x if the spectrum of x is an interval of integers. A proper edge t-coloring of a graph G is persistent-interval for its vertex x if the spectrum of x is an interval of integers beginning from the color 1. For graphs G from some classes of graphs, we obtain estimates for the possible number of vertices for which a proper edge t-coloring of G can be interval or persistent-interval.

1. Introduction We consider undirected, simple, finite, connected graphs. For a graph G, we denote by V (G) and E(G) the sets of its vertices and edges, respectively. For any x ∈ V (G), dG (x) denotes the degree of the vertex x in G. For a graph G, we denote by ∆(G) the maximum degree of a vertex of G. A function ϕ : E(G) → {1, 2, . . . , t} is called a proper edge t-coloring of a graph G if each of t colors is used, and adjacent edges are colored differently. The set of all proper edge t-colorings of G is denoted by α(G, t). The minimum value of t for which there exists a proper edge t-coloring of a graph G is called a chromatic index [ 22] of G and is denoted by χ′ (G). Let us also define the set α(G) of all proper edge colorings of the graph G |E(G)|

α(G) ≡

[

α(G, t).

t=χ′ (G)

If G is a graph, ϕ ∈ α(G), x ∈ V (G), then the set of colors of edges of G incident with x is called a spectrum of the vertex x in the coloring ϕ of the graph G and is denoted by SG (x, ϕ). An arbitrary nonempty subset of consecutive integers is called an interval. An interval with the minimum element p and the maximum element q is denoted by [p, q]. An interval D is called a h-interval if |D| = h. ∗

email: [email protected]

2

R.R. Kamalian

For any real number ξ, we denote by ⌊ξ⌋ (⌈ξ⌉) the maximum (minimum) integer which is less (greater) than or equal to ξ. If G is a graph, ϕ ∈ α(G), and x ∈ V (G), then we say that ϕ is interval (persistentinterval) for x if SG (x, ϕ) is a dG (x)-interval (a dG (x)-interval with 1 as its minimum element). For an arbitrary graph G and any ϕ ∈ α(G), we denote by fG,i (ϕ)(fG,pi (ϕ)) the number of vertices of the graph G for which ϕ is interval (persistent-interval). For any graph G, let us [ 17] set ηi (G) ≡ max fG,i (ϕ), ϕ∈α(G)

ηpi (G) ≡ max fG,pi (ϕ). ϕ∈α(G)

A bipartite graph G with bipartition (X, Y ) is called (a, b)-biregular, if dG (x) = a for any vertex x ∈ X, and dG (y) = b for any vertex y ∈ Y . The terms and concepts that we do not define can be found in [ 23]. It is clear that if for any graph G ηpi (G) = |V (G)|, then χ′ (G) = ∆(G). For a regular graph G, these two conditions are equivalent: ηpi (G) = |V (G)| iff χ′ (G) = ∆(G). It is known [ 15, 19] that for a regular graph G, the problem of deciding whether χ′ (G) = ∆(G) or not is NP -complete. It means that for a regular graph G, the problem of deciding whether ηpi (G) = |V (G)| or not is also NP -complete. For any tree G, some necessary and sufficient condition for ηpi (G) = |V (G)| was obtained in [ 8]. In this paper, for an arbitrary regular graph G, we obtain a lower bound for the parameter ηpi (G). If G is a graph, R0 ⊆ V (G), and the coloring ϕ ∈ α(G) is interval (persistent-interval) for any x ∈ R0 , then we say that ϕ is interval (persistent-interval) on R0 . ϕ ∈ α(G) is called an interval coloring of a graph G if ϕ is interval on V (G). We define the set N as the set of all graphs for which there is an interval coloring. Clearly, for any graph G, G ∈ N if and only if ηi (G) = |V (G)|. The notion of an interval coloring was introduced in [ 6]. In [ 6, 16, 7] it is shown that if G ∈ N, then χ′ (G) = ∆(G). For a regular graph G, these two conditions are equivalent: G ∈ N iff χ′ (G) = ∆(G) [ 6, 16, 7]. Consequently, for a regular graph G, four conditions are equivalent: G ∈ N, χ′ (G) = ∆(G), ηi (G) = |V (G)|, ηpi (G) = |V (G)|. It means that for any regular graph G, 1. the problem of deciding whether or not G has an interval coloring is NP -complete, 2. the problem of deciding whether ηi (G) = |V (G)| or not is NP -complete. In this paper, for an arbitrary regular graph G, we obtain a lower bound for the parameter ηi (G). We also obtain some results for bipartite graphs. The complexity of the problem of existence of an interval coloring for bipartite graphs is investigated in [ 3, 9, 21]. In [ 16] it is shown that for a bipartite graph G with bipartition (X, Y ) and ∆(G) = 3 the problem of existence of a proper edge 3-coloring which is persistent-interval on X ∪ Y (or even only on Y [ 6, 16]) is NP -complete. Suppose that G is an arbitrary bipartite graph with bipartition (X, Y ). Then ηi (G) ≥ max{|X|, |Y |}. Suppose that G is a bipartite graph with bipartition (X, Y ) for which there exists a coloring ϕ ∈ α(G) persistent-interval on Y . Then ηpi (G) ≥ 1 + |Y |.

Estimates for the number of vertices with an interval spectrum in proper edge colorings of some graphs3 Some attention is devoted to (a, b)-biregular bipartite graphs [ 4, 14, 13, 18] in the case b = a + 1. We show that if G is a (k − 1, k)-biregular bipartite graph, k ≥ 4, then & ' k−1 k ηi (G) ≥ · |V (G)| +  k  · |V (G)| . 2k − 1 · (2k − 1) 2 We show that if G is a (k − 1, k)-biregular bipartite graph, k ≥ 3, then ηpi (G) ≥

k · |V (G)|. 2k − 1

2. Results Theorem 1 [ 17] If G is a regular graph with χ′ (G) = 1 + ∆(G), then ' & |V (G)| . ηpi (G) ≥ 1 + ∆(G) Proof. Suppose that β ∈ α(G, 1 + ∆(G)). For any j ∈ [1, 1 + ∆(G)], define VG,β,j ≡ {x ∈ V (G)/j 6∈ SG (x, β)}. For arbitrary integers j ′ , j ′′ , where 1 ≤ j ′ < j ′′ ≤ 1 + ∆(G), we have VG,β,j ′ ∩ VG,β,j ′′ = ∅ and

1+∆(G)

[

VG,β,j = V (G).

j=1

Hence, there exists j0 ∈ [1, 1 + ∆(G)] for which & ' |V (G)| |VG,β,j0 | ≥ . 1 + ∆(G) Set R0 ≡ VG,β,j0 . Case 1 1j0 = 1 + ∆(G). Clearly, β is persistent-interval on R0 . Case 2 2 j0 ∈ [1, ∆(G)]. Define a function ϕ : E(G) → [1, 1 + ∆(G)]. For any e ∈ E(G), set:  if β(e) 6∈ {j0 , 1 + ∆(G)}  β(e), ϕ(e) ≡ j0 , if β(e) = 1 + ∆(G)  1 + ∆(G), if β(e) = j0 .

It is not difficult to see that ϕ ∈ α(G, 1 + ∆(G)) and ϕ is persistent-interval on R0 .

4

R.R. Kamalian

′ Corollary 1 [ 17] If G is a cubic graph, m there exists a coloring from α(G, χ (G)) l then vertices of G. which is persistent-interval for at least |V (G)| 4

Theorem 2 [ 17] If G is a regular graph with χ′ (G) = 1 + ∆(G), then & ' |V (G)| ηi (G) ≥  1+∆(G)  . 2

Proof. Suppose that β ∈ α(G, 1 + ∆(G)). For any j ∈ [1, 1 + ∆(G)], define VG,β,j ≡ {x ∈ V (G)/j 6∈ SG (x, β)}. For arbitrary integers j ′ , j ′′ , where 1 ≤ j ′ < j ′′ ≤ 1 + ∆(G), we have VG,β,j ′ ∩ VG,β,j ′′ = ∅ and

1+∆(G)

[

VG,β,j = V (G).

j=1

 1+∆(G)  ], let us define the 2    VG,β,2i−1 ∪ VG,β,2i , V (G, β, i) ≡   V G,β,1+∆(G) ,

For any i ∈ [1,

subset V (G, β, i) of the set V (G) as follows: ] if ∆(G) is odd and i ∈ [1, 1+∆(G) 2 ∆(G) or ∆(G) is even and i ∈ [1, 2 ], if ∆(G) is even and i = 1 + ∆(G) . 2

For arbitrary integers i′ , i′′ , where 1 ≤ i′ < i′′ ≤

 1+∆(G)  , we have 2

V (G, β, i′ ) ∩ V (G, β, i′′ ) = ∅ and

 1+∆(G)  2 [

V (G, β, i) = V (G).

i=1

   for which Hence, there exists i0 ∈ 1, 1+∆(G) 2 & ' |V (G)| |V (G, β, i0)| ≥  1+∆(G)  . 2

Set R0 ≡ V (G, β, i0 ). Case 3 1 i0 =

 1+∆(G)  . 2

Case 4 1.a ∆(G) is even. Clearly, β is interval on R0 .

Estimates for the number of vertices with an interval spectrum in proper edge colorings of some graphs5 Case 5 1.b ∆(G) is odd. Define a function ϕ : E(G) → [1, 1 + ∆(G)]. For any e ∈ E(G), set:  (β(e) + 1)(mod(1 + ∆(G))), if β(e) 6= ∆(G), ϕ(e) ≡ 1 + ∆(G), if β(e) = ∆(G). It is not difficult to see that ϕ ∈ α(G, 1 + ∆(G)) and ϕ is interval on R0 .   Case 6 2 1 ≤ i0 ≤ ∆(G)−1 . 2 Define a function ϕ : E(G) → [1, 1 + ∆(G)]. For any e ∈ E(G), set:  (β(e) + 2 + ∆(G) − 2i0 )(mod(1 + ∆(G))), if β(e) 6= 2i0 − 1, ϕ(e) ≡ 1 + ∆(G), if β(e) = 2i0 − 1. It is not difficult to see that ϕ ∈ α(G, 1 + ∆(G)) and ϕ is interval on R0 . Corollary 2 [ 17] If G is a cubic graph, then there exists a coloring from α(G, χ′(G)) vertices of G. which is interval for at least |V (G)| 2 Theorem 3 [ 6, 16, 7] Let G be a bipartite graph with bipartition (X, Y ). Then there exists a coloring ϕ ∈ α(G, |E(G)|) which is interval on X. Corollary 3 Let G be a bipartite graph with bipartition (X, Y ). Then ηi (G) ≥ max{|X|, |Y |}. Theorem 4 [ 1, 6, 7] Let G be a bipartite graph with bipartition (X, Y ) where dG (x) ≤ dG (y) for each edge (x, y) ∈ E(G) with x ∈ X and y ∈ Y . Then there exists a coloring ϕ0 ∈ α(G, ∆(G)) which is persistent-interval on Y . Theorem 5 Suppose G is a bipartite graph with bipartition (X, Y ), and there exists a coloring ϕ0 ∈ α(G, ∆(G)) which is persistent-interval on Y . Then, for an arbitrary vertex x0 ∈ X, there exists ψ ∈ α(G, ∆(G)) which is persistent-interval on {x0 } ∪ Y . Proof. Case 7 1 SG (x0 , ϕ0 ) = [1, dG (x0 )]. In this case ψ is ϕ0 . Case 8 2 SG (x0 , ϕ0 ) 6= [1, dG (x0 )]. Clearly, [1, dG (x0 )]\SG (x0 , ϕ0 ) 6= ∅, SG (x0 , ϕ0 )\[1, dG (x0 )] 6= ∅. Since |SG (x0 , ϕ0 )| = |[1, dG (x0 )]| = dG (x0 ), there exists ν0 ∈ [1, dG (x0 )] satisfying the condition |[1, dG (x0 )]\SG (x0 , ϕ0 )| = |SG (x0 , ϕ0 )\[1, dG (x0 )]| = ν0 . Now let us construct the sequence Θ0 , Θ1 , . . . , Θν0 of proper edge ∆(G)-colorings of the graph G, where for any i ∈ [0, ν0 ], Θi is persistent-interval on Y . Set Θ0 ≡ ϕ0 . Suppose that for some k ∈ [0, ν0 − 1], the subsequence Θ0 , Θ1 , . . . , Θk is already constructed. Let tk ≡ max(SG (x0 , Θk )\[1, dG (x0 )]), sk ≡ min([1, dG (x0 )]\SG (x0 , Θk )). Clearly, tk > sk . Consider the path P (k) in the graph G of maximum length with the initial vertex x0 whose edges are alternatively colored by the colors tk and sk . Let Θk+1 is obtained from Θk by interchanging the two colors tk and sk along P (k). It is not difficult to see that Θν0 is persistent-interval on {x0 } ∪ Y . Set ψ ≡ Θν0 .

6

R.R. Kamalian

Corollary 4 Let G be a bipartite graph with bipartition (X, Y ) where dG (x) ≤ dG (y) for each edge (x, y) ∈ E(G) with x ∈ X and y ∈ Y . Let x0 be an arbitrary vertex of X. Then there exists a coloring ϕ0 ∈ α(G, ∆(G)) which is persistent-interval on {x0 } ∪ Y . Corollary 5 [ 17] Let G be a bipartite graph with bipartition (X, Y ) where dG (x) ≤ dG (y) for each edge (x, y) ∈ E(G) with x ∈ X and y ∈ Y . Then ηpi (G) ≥ 1 + |Y |. Remark 1 Notice that the complete bipartite graph Kn+1,n for an arbitrary positive integer n satisfies the conditions of Corollary 5. Is is not difficult to see that ηpi (Kn+1,n ) = 1 + n. It means that the bound obtained in Corollary 5 is sharp since in this case |Y | = n. Remark 2 Let G be a bipartite (k − 1, k)-biregular graph with bipartition (X, Y ), where |Y | (G)| k ≥ 3. Then the numbers |X| , k−1 , and |V2k−1 are integer. It follows from the equalities k gcd(k − 1, k) = 1 and |E(G)| = |X| · (k − 1) = |Y | · k. Theorem 6 [ 17] Let G be a bipartite (k − 1, k)-biregular graph, where k ≥ 4. Then & ' k k−1 · |V (G)| . · |V (G)| +  k  ηi (G) ≥ 2k − 1 · (2k − 1) 2

Proof. Suppose that (X, Y ) is a bipartition of G. Clearly, χ′ (G) = ∆(G) = k. Suppose that β ∈ α(G, k). For any j ∈ [1, k], define: VG,β,j ≡ {x ∈ X/j 6∈ SG (x, β)}. For arbitrary integers j ′ , j ′′ , where 1 ≤ j ′ < j ′′ ≤ k, we have VG,β,j ′ ∩ VG,β,j ′′ = ∅ and k [

VG,β,j = X.

j=1

For any i ∈ [1, ⌈ k2 ⌉], let us define the subset V (G, β, i) of the set X as follows:  ]  VG,β,2i−1 ∪ VG,β,2i , if k is odd and i ∈ [1, k−1 2 k or k is even and i ∈ [1, 2 ], V (G, β, i) ≡  . VG,β,k , if k is odd and i = 1+k 2   For arbitrary integers i′ , i′′ , where 1 ≤ i′ < i′′ ≤ k2 , we have V (G, β, i′ ) ∩ V (G, β, i′′ ) = ∅

and

  k 2 [ i=1

V (G, β, i) = X.

Estimates for the number of vertices with an interval spectrum in proper edge colorings of some graphs7    Hence, there exists i0 ∈ 1, k2 for which |V (G, β, i0 )| ≥ Set R0 ≡ Y ∪ V (G, β, i0 ). It is not difficult to verify that

Case 9 1 i0 =

&

' |X| k . 2

& ' k k−1 · |V (G)| +  k  |R0 | ≥ · |V (G)| . 2k − 1 · (2k − 1) 2

k . 2

Case 10 1.a k is odd. Clearly, β is interval on R0 . Case 11 1.b k is even. Define a function ϕ : E(G) → [1, k]. For any e ∈ E(G), set:  (β(e) + 1)(modk), if β(e) 6= k − 1, ϕ(e) ≡ k, if β(e) = k − 1. It is not difficult to see that ϕ ∈ α(G, k) and ϕ is interval on R0 .     Case 12 2 i0 ∈ 1, k2 − 1 . Define a function ϕ : E(G) → [1, k]. For any e ∈ E(G), set:  (β(e) + 1 + k − 2i0 )(modk), if β(e) 6= 2i0 − 1, ϕ(e) ≡ k, if β(e) = 2i0 − 1. It is not difficult to see that ϕ ∈ α(G, k) and ϕ is interval on R0 . Corollary 6 [ 17] Let G be a bipartite (k − 1, k)-biregular graph, where k is even and k ≥ 4. Then k+1 ηi (G) ≥ · |V (G)|. 2k − 1 Corollary 7 [ 17] Let G be a bipartite (3, 4)-biregular graph. Then there exists a coloring from α(G, 4) which is interval for at least 57 |V (G)| vertices of G. Remark 3 For an arbitrary bipartite graph G with ∆(G) ≤ 3, there exists an interval coloring of G [ 12, 10, 11]. Consequently, if G is a bipartite (2, 3)-biregular graph, then ηi (G) = |V (G)|. Remark 4 Some sufficient conditions for existence of an interval coloring of a (3, 4)biregular bipartite graph were obtained in [ 2, 5, 20].

8

R.R. Kamalian

Theorem 7 [ 17] Let G be a bipartite (k − 1, k)-biregular graph, where k ≥ 3. Then k · |V (G)|. 2k − 1

ηpi (G) ≥

Proof. Suppose that (X, Y ) is a bipartition of G. Clearly, χ′ (G) = ∆(G) = k. Suppose that β ∈ α(G, k). For any j ∈ [1, k], define: VG,β,j ≡ {x ∈ X/j 6∈ SG (x, β)}. For arbitrary integers j ′ , j ′′ , where 1 ≤ j ′ < j ′′ ≤ k, we have VG,β,j ′ ∩ VG,β,j ′′ = ∅ and

k [

VG,β,j = X.

j=1

Hence, there exists j0 ∈ [1, k] for which |VG,β,j0 | ≥

|X| . k

Set R0 ≡ Y ∪ VG,β,j0 . It is not difficult to verify that |R0 | ≥

k · |V (G)|. 2k − 1

Case 13 1 j0 = k. Clearly, β is persistent-interval on R0 . Case 14 2 j0 ∈ [1, k − 1]. Define a function ϕ : E(G) → [1, k]. For   β(e), j0 , ϕ(e) ≡  k,

any e ∈ E(G), set: if β(e) 6∈ {j0 , k} if β(e) = k if β(e) = j0 .

It is not difficult to see that ϕ ∈ α(G, k) and ϕ is persistent-interval on R0 .

Corollary 8 [ 17] Let G be a bipartite (3, 4)-biregular graph. Then there exists a coloring from α(G, 4) which is persistent-interval for at least 74 |V (G)| vertices of G. 3. Acknowledgment The author thanks professors A.S. Asratian and P.A. Petrosyan for their attention to this work.

Estimates for the number of vertices with an interval spectrum in proper edge colorings of some graphs9 REFERENCES 1. A.S. Asratian, Investigation of some mathematical model of Scheduling Theory, Doctoral dissertation, Moscow University, 1980 (in Russian). 2. A.S. Asratian, C.J. Casselgren, A sufficient condition for interval edge colorings of (4, 3)-biregular bipartite graphs, Research report LiTH-MAT-R-2006-07, Link¨oping University, 2006. 3. A.S. Asratian, C.J. Casselgren, Some results on interval edge colorings of (α, β)biregular bipartite graphs, Research report LiTH-MAT-R-2006-09, Link¨oping University, 2006. 4. A.S. Asratian, C.J. Casselgren, On interval edge colorings of (α, β)-biregular bipartite graphs, Discrete Math 307 (2007), pp. 1951–1956. 5. A.S. Asratian, C.J. Casselgren, J. Vandenbussche, D.B. West, Proper path-factors and interval edge-coloring of (3, 4)-biregular bigraphs, J. of Graph Theory 61 (2009), pp. 88–97. 6. A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987), Yerevan State University, pp. 25–34 (in Russian). 7. A.S. Asratian, R.R. Kamalian, Investigation of interval edge-colorings of graphs, Journal of Combinatorial Theory. Series B 62 (1994), no.1, pp. 34–43. 8. Y. Caro, J. Sch¨onheim, Generalized 1-factorization of trees, Discrete Math 33 (1981), pp. 319–321. 9. K. Giaro, The complexity of consecutive ∆-coloring of bipartite graphs: 4 is easy, 5 is hard. Ars Combin. 47(1997), 287–298. 10. K. Giaro, Compact Task Scheduling on Dedicated Processors with no Waiting Periods , Ph.D. Thesis, Technical University of Gda´ nsk, ETI Faculty, Gda´ nsk (1999) (in Polish). 11. K. Giaro, M. Kubale, M. Malafiejski, Compact scheduling in open shop with zero-one time operations, INFOR 37 (1999), pp. 37–47. 12. H. Hansen, Scheduling with minimum waiting periods , Master Thesis, Odense University, Odense, Denmark, 1992 (in Danish). 13. D. Hanson, C.O.M. Loten, A lower bound for Interval colouring bi-regular bipartite graphs, Bulletin of the ICA 18 (1996), pp. 69–74. 14. D. Hanson, C.O.M. Loten, B. Toft, On interval colourings of bi-regular bipartite graphs, Ars Combin. 50 (1998), pp. 23–32. 15. I. Holyer, The NP -completeness of edge-coloring, SIAM J. Comput. 10 (1981), pp. 718–720. 16. R.R. Kamalian, Interval Edge Colorings of Graphs, Doctoral dissertation, the Institute of Mathematics of the Siberian Branch of the Academy of Sciences of USSR, Novosibirsk, 1990 (in Russian). 17. R.R. Kamalian, On a number of vertices with an interval spectrum in proper edge colorings of some graphs. Research report LiTH-MAT-R-2011/03-SE, Link¨oping University, 2011. 18. A.V. Kostochka, Unpublished manuscript, 1995. 19. D. Leven and Z. Galil, NP -completeness of finding the chromatic index of regular graphs, J. Algorithms 4 (1983), pp. 35–44. 20. A.V. Pyatkin, Interval coloring of (3, 4)-biregular bipartite graphs having large cubic

10

R.R. Kamalian

subgraphs, J. of Graph Theory 47 (2004), pp. 122–128. 21. S.V. Sevast’janov, Interval colorability of the edges of a bipartite graph, Metody Diskret. Analiza 50(1990), 61–72 (in Russian). 22. V.G. Vizing, The chromatic index of a multigraph, Kibernetika 3 (1965), pp. 29–39. 23. D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 1996.