Interval edge-colorings of graph products

Interval edge-colorings of graph products Petros Petrosyan Institute for Informatics and Automation Problems of NAS of RA, Department of Informatics and Applied Mathematics, YSU, Yerevan, Arm enia e-mail: pet_petros@ip ia.sci.am Hrant Khachatrian, Liana Yepremyan Department of Informatics and Applied Mathematics, YSU, Yerevan, Arm enia e-mails: hrant@ege rn.net, liana.yepremyan2009@gmail.com Hovhannes Tananyan Department of Applied Mathematics and Inform atics, RAU, Yerevan, Arm enia e-mail: HTananyan@y ahoo.com ABSTRACT An interval t coloring of a graph is a proper edge- coloring of with colors 1, such that at least one edge of G is colored b y and the edges incident to each vertex − G G 2 , , t … , 1 ,2 , , , ii t = … () vV G ∈ are colored by consecutive colors, where is the degree of the vertex in . In this paper interval edge-colorings of various graph products are investigated. ( ) G dv ( ) G dv v G Keywords Edge-coloring, interval coloring, products of graphs, regular graph 1. INTRODUCTION All graphs considered in this paper are finite, und irected, connected and have no loops or multiple edges. Let and denote the sets of vertices and edges of a graph , respectivel y. The degree of a vertex () VG () EG G () vV G ∈ is denoted by , the maximum degree of a vert ex in by , the chromatic index of G by ( ) G dv G () G Δ () G χ ′ , and the diameter of G by . We use the standard notations and for the simple path, simple c ycle, complete graph on n vertices and the dimensional cube, respectively. () dG ,, nn n CK P n Q n − Let and G H be two graphs. The Cartesian product is d efined as follows: GH  () ( ) , () VG V H VG H =×  () ( ) { } 11 2 2 1 2 1 2 12 1 2 ,, and ( ) or and ( ) () . uv u v u u v v E H vv u u E G EG H == ∈ =∈  The tensor (direct) product G is defined as follows: H × () ( ) , () VG V H VG H =× × () ( ) { } 11 2 2 1 2 1 2 ,, ( ) and ( ) () . uv u v u u E G v v EH EG H =∈ ∈ × The strong tensor (semistrong) product GH ⊗ is defined as follows: () ( ) , () VG V H VG H =× ⊗ () ( ) { } 11 2 2 1 2 1 2 12 1 2 ,, ( ) and ( ) or and ( ) () . uv u v u u E G v v EH vv u u E G EG H =∈ ∈ =∈ ⊗ The strong product GH × is defined as follows: () ( ) , () VG V H VG H ×= × () ( ) { } 11 2 2 1 2 1 2 12 1 2 1 2 1 2 ,, ( ) and ( ) or and ( ) or and ( ) () . uv u v u u E G v v EH uu v v E H vv u u E G EG H ×= ∈ ∈ =∈ = ∈ The lexicographic product ( composition) [ ] H G is defined as follows: [] ( ) () ( ) , HV G V H VG =× [] ( ) () ( ) { } 11 2 2 1 2 1 2 12 ,, () o r and ( ) . Hu v u v u u E G u vv E H EG u = ∈= ∈ The terms and concepts that w e do not define can be found in [3,6,7,12] . An interval t − coloring [1] of a graph is a proper edge-coloring of with colors 1, such that at least one edge of G is colored by and the edges incident to each vertex are colored by consecutive colors. G G 2 , , t … , 1 ,2 , , , ii t = … () vV G ∈ ( ) G dv For let 1, t ≥ t N denote the set of graphs which have an interval t − coloring, and assume: 1 t t ≥ ≡ ∪ N N . For a graph , G ∈ N the least and the greatest values of for which t t G ∈ N are denoted by and , respectively . () wG () WG In [1,2], Asratian and Kamalian proved the following theorem. Theorem 1 . Let be a regular graph. Then G 1. G ∈ N if and only if () () GG χ ′ =Δ . 2. If G ∈ N and () () Gt W G Δ ≤≤ , then t G ∈ N . Later, they derived some upper bounds for depending on degrees and diameter of a connected graph () WG G ∈ N . Theorem 2 . [2] If G is a connected graph and , G ∈ N then ( ) ( ) () () 1 1 . () dG G WG 1 + Δ− + ≤ Moreover, if is also bipartite, then G ( ) () 1 1 () () G WG d G Δ− + ≤ . In [9], Petrosyan investigated interval edge-co lorings of complete graphs and n − dimensional cubes. In particular, he proved the following two th eorems. Theorem 3 . If where is odd and is nonnegative, then , 2 q np = p q ( ) 2 42 n . K np W ≥− − − q Theorem 4 . For any , n ∈  () () 1 . 2 n nn Q W + ≥ In this paper interval edge-colo rings of various graph products are investigated. 2. INTERVAL EDGE-COLORINGS OF CARTESIAN PRODUCTS OF GRAPHS First, interval edge -colorings of Cartesian products of graphs were investigated by Gi aro and Kubale in [4], where they proved the fo llowing Theorem 5. If then , G ∈ N ( m m GP ) ∈ ∈  N and () 2 2. n n GC ≥ ∈ N In the same paper the y proved the following theorem. Theorem 6. If ( ) 12 k i nn n n GP P P ∈ =     or or then and () 2 ,2 mn mn GP C ∈≥ =  ( 22 ,2 , mn mn GC C ≥ = ) ) ) , ) G ∈ N () () . wG G =Δ For of these graphs, Petrosyan, Karapetyan [8] and Petrosyan, Khachatrian [11] proved the following () WG Theorem 7. If , then , and if then and if then If then ( 2 ,2 mn mn GP C ∈≥ =  () 3 2 WG m n ≥+ − () 22 ,2 , mn mn GP C ∈≥ =  () 4 2 2 , WG m n ≥+ − ( 22 1 , mn mn GP C + ∈ =  () 4 2 1 . WG m n ≥+ − ( 22 ,2 , mn mn GC C ≥ = { } 32 , 32 () m a x , mn nm WG ++ + + ≥ and if then () 22 1 2, , mn mn GC C + ≥∈ =  2 2 3, i f is even, 2 2 2, if is odd. () mn m mn m WG ⎧ ++ ⎪ ⎨ ++ ⎪ ⎩ ≥ In [5,7], Giaro and Kubale proved the following theorem . Theorem 8. If then and ,, GH ∈ N GH ∈ N ( ) ( ) () ( ) , () ( ) . GH w G w H GH W G W H wW ≤ + ≥ + We improve the lower bound in Theorem 8 for ( ) GH W  when , GH ∈ N and H is an r − regular graph. More precisely, we s how that the following theorem holds. Theorem 9. If and , GH ∈ N H is an regular graph , then GH and r − ∈ N ( ) () ( ) . GH W G W H r W ≥ + + Corollary 1. If is an regular graph, G r − H is an regular graph and then r ′ − ,, GH ∈ N GH ∈ N and ( ) { } () ( ) m a x , . GH W G W H r r W ′ ≥ + + Corollary 2. If is an regular graph, i G i r − , i G ∈ N and then 1, in ≤≤ 12 , n rr r ≥ ≥≥  12 n GG G   ∈  N and () 1 12 11 1 () . nn k ni ik i GG G W G r W − == =   ≥ + ∑∑ i ∑  Next, we consider Hamming graphs. Recall that the Hamming graph ( ) 12 ,, , , , 1 n i mm m m , H in ∈ ≤ ≤ … is the Cartesian product of complete graphs 12 . n mm m K KK    In [10], Petrosyan noted that ( ) 12 ,, , n mm m H ∈ … N if and only if is even. Moreover, he proved the following 12 n mm m  Theorem 10. If where is odd and is nonnegative, then , 2 q mp = p q ( ) 2, 2, , 2 mm m H ∈ … N and () ( ) () 2, 2, , 2 2 1 , mm m m n wH =− … () ( ) () 2, 2, , 2 4 2 . mm m m p q n WH ≥− − − … B y Theorem 3 and Corollary 2, we can improve the lower bound in Theorem 10. Theorem 11. If where is odd and is nonnegative, 1, , 2 i ii q mp = i p i q in ≤ ≤ then ( ) 12 2, 2 , , 2 n mm m H ∈ … N and () () () 12 1 2, 2 , , 2 2 1 , n ni i mm m m wH = =− ∑ … () () () ( 1 12 11 2, 2 , , 2 4 2 2 1 . nn ni i i n ii mm m m p q i m WH − − == ≥− − − + ⋅ − ∑∑ … ) i Corollary 3. If where is odd and is nonnegative, then , 2 q mp = p q ( ) 2, 2, , 2 mm m H ∈ … N and () ( ) () 2, 2, , 2 2 1 , mm m m n wH =− … () () () ( ) ( ) 12 1 2, 2 , , 2 4 2 . 2 nn m mm m m p q n WH − − ≥− − − ⋅ + … Note that Corollary 3 generalizes Theorem 4, since () 2,2, ,2 . n Q H = … Also, we pro vide some sufficient conditions for ( ) () ( ) () HW G W H d G WG ≥+ +  r ⋅ whe n , GH ∈ N and H is an r − regular graph. In particular, we prove the following two theorems. Theorem 12. If is an regular graph and G r − , G ∈ N then and ( 2 2 n n GC ≥ ∈ N ) ( ) 22 () ( ) . nn WG W n r WG C C ≥+ + ⋅  Theorem 13. If is an regular graph and G r − , G ∈ N then ( m m GP ) ∈ ∈  N and ( ) () () ( ) 1 . mm WG W m r WG P P ≥+ + − ⋅  Corollary 4. If where , 2 q np = p is odd and is nonnegative, then q ( ) 22 2 24 1 nn . K Cn n p W ≥ + − − − q Corollary 5. If G is an regular graph and r − , G ∈ N then () n n GQ ∈ ∈  N and () ( ) 21 () . 2 n nn r QW G WG ++ ≥+  Note that the lower bound in Corollary 4 is clos e to the upper bound for ( ) 22 , nn KC W  since and () 22 21 nn KC n = + Δ ( ) 22 1, nn KC n d = + by Theorem 2, we have ( ) 22 2 24 nn KC n n W ≤ + + 1 . We also confirm the conjectur e on the n − dimensional cube [9] and show that n Q () ( ) 1 2 n nn Q W + = for any . n ∈  Next, we obtain some partial results for the case w hen one of the factors has no interval coloring. Theorem 14 . Fo r any and , n m + ∈∈  () () 21 2 21 nn KP m n m n W + ≥ + + − , ) () () ( 21 41 . 2 nm mn m KQ W + ++ ≥ Corollary 6. For any , n ∈  () 21 2 55 . 2 nn nn KQ W + + ≥ Note that the lower bound in Corollary 6 is clos e to the upper bound for ( ) 21 , nn KQ W +  since ( ) 21 3 nn K Qn + = Δ and ( ) 21 1, nn KQn d + = + by Theorem 2, we have ( ) 21 2 35 nn KQ n n W + ≤ + − 1 . 3. INTERVAL EDGE-COLORINGS OF TENSOR PRODUCTS OF GRAPHS First, interval edge-colori ngs of tensor products of graphs were considered by Giaro and Kubale in [7], where they noted that there a re such that , GH ∈ N . GH × ∉ N On the other hand, Petrosyan [10] proved that if one of the factors belongs to and the other is regular, then N . GH ×∈ N Theorem 15. If and G ∈ N H is an regular graph, then Moreover, r − . GH ×∈ N () () GH ww G ×≤ r ⋅ and ( ) () . GH WW G ×≥ ⋅ r In the same paper the author formulated th e following Problem 1. Are there gra phs , GH ∉ N such that ? GH ×∈ N In [13], Yepremyan constructed graphs , GH ∉ N such that If we take the Sylvester g raph S as and the triangle as . GH ×∈ N G 3 C H , then 3 . SC ×∈ N 4. INTERVAL EDGE-COLORINGS OF STRONG TENSOR PRODUCTS O F GRAPHS First, interval edge-colorings of strong tensor products of graphs were considered by Petrosyan in [10], where he proved that if one of the factors belongs to and the other is regular, then N . GH ⊗∈ N Theorem 16. If and G ∈ N H is an regular graph, then Moreover, r − . GH ⊗∈ N ( ) ( ) 1 () GH r ww G ⊗≤ + and () ( 1 () . GH r WW G ⊗≥ + ) In the same paper the author formulated th e following Problem 2. Are there gra phs , GH ∉ N such that ? GH ⊗∈ N In [13], Yepremyan constructed graphs , GH ∉ N such that If we take the Sylvester graph as and the triangle as . GH ⊗∈ N S G 3 C H , then 3 . SC ⊗∈ N 5. INTERVAL EDGE-COLORINGS OF STRONG PRODUCTS OF GRAPHS First, interval edge-colori ngs of strong prod ucts of graphs were considered by Giaro and Kubale [7], where they noted that there are such that , GH ∈ N . GH × ∉ N On the other hand, Petros yan [10] proved that if factors belong to N and one of them is regular, then . GH × ∈ N Theorem 17. If , GH ∈ N and H is an r − regular graph, then . GH × ∈ N Moreover, ( ) () 1 () GH r r ww G ×≤ + + and ( ) () 1 () . GH r r WW G × ≥+ + Note that there are graphs and G H for which H G × ∈ N , but For example, , GH ∈∉ N . N 23 , C K × ∈ N but Moreover, in [10], Petrosyan noted that if G and 3 . C ∉ N H are regular graphs and one of them belongs to N , then . GH × ∈ N In the same paper the author formulated the following Problem 3. Are there gra phs , GH ∉ N such that ? GH × ∈ N This problem is still open. 6. INTERVAL EDGE-COLORINGS OF LEXICOGRAPHIC PRODUCTS OF GRAPHS First, interval edge-colori ngs of lexicographic products of graphs were considered by Giaro and Kubale in [7], where they posed the follow ing Problem 4. Does [ ] H G ∈ N if ,? GH ∈ N In [10], Petrosyan proved the following two results. Theorem 18. If , G ∈ N then for any 1 nK G ⎡⎤ ⎣⎦ ∈ N . n ∈  Moreover, [] ( ) 1 () Gn K ww G ≤ ⋅ n and [] ( ) ( ) 1 () 1 1. Gn K W G Wn ≥+ ⋅− Theorem 19. If , GH ∈ N and H is an r − regular graph, then [ ] . H G ∈ N Moreover, if () , VH n = [] ( ) [] ( ) and . () () GH r GH r w w Gn W W Gn ≤ +≥ ⋅⋅ + For some cases, the lower bound in Theorem 19 was improved by Yepremyan in [13] . Theorem 20. If H ∈ N and H is an r − regular graph, then for any , n ∈  [ ] n PH ∈ N and 1. [] ( ) [] ( ) , nn PH PH w =Δ 2. [] ( ) ( ) () . () n n PH P VH r r WW ≥+ ⋅ + Theorem 21. If H ∈ N and H is an r − regular graph, then for any 2, n ≥ [ ] 2 n CH ∈ N and 1. [] ( ) [] ( ) 22 , nn CH CH w =Δ 2. [] ( ) ( ) ( ) ( ) 22 1( ) nn CH C V H r WW ≥− ⋅ . + Theorem 22. If H ∉ N and H is an r − regular graph, then for any , n ∈  [ ] 2 n PH ∈ N and 1. [] ( ) [] ( ) 22 , nn PH PH w =Δ 2. [] ( ) 2 2 () ( ) . n n PH W P V H n r W ≥⋅ + ⋅ Theorem 23. If and H ∉ N H is an regular graph, then for any r − 2, n ≥ [ ] 2 n CH ∈ N and 1. [] ( ) [] ( ) 22 , nn CH CH w =Δ 2. [] () () () 22 1( ) 2 nn n CH C V H r WW . ⎡ ⎤ ≥− + ⋅ ⎢ ⎥ ⎢ ⎥ ⋅ Also, Yepremyan [13] noted that there are graphs such that , GH ∉ N [ ] . H G ∈ N If we take as G the triangle and as 3 C H the Petersen graph , then P 3 . P C ⎡⎤ ⎢⎥ ⎣⎦ ∈ N Finally, she proved [13] that if T is a tree, then and where is a star n P T ⎡⎤ ⎣⎦ ∈ N , n S T ⎡⎤ ⎣⎦ ∈ N n S 1, . n K Before we formulate these r esu lts we need some definitions. Let be a tree and T { } ,, , 12 , 2 . () n vv v n VT ≥ = … Let ( ) , ij vv P be a simple path joining and i v , j v ( ) , ij vv VP and ( ) , ij vv EP denote the sets of verti ces and edges of the path, respectively. For a simple path ( ) , ij vv P , define ( ) , ij vv L as follows: () () () { } { () } ,, , , | ( ) , \ , . ij ij ij ij ij vv E P vv u vu v ET u V Pvv vv vV P v v L =+ ∈ ∈ ∉ , Let be a center of and () CT , T () F T be a set of pendant vertices of . T Define: ( ) () () max max , () uC T v F T L uv mT ∈∈ = and () ,( ) max , . () uv F T L uv MT ∈ = Now we can present these r esults . Theorem 24. If is a tree, then for any T , n ∈  and n P T ⎡⎤ ⎣⎦ ∈ N 1. [] ( ) ( ) () () 1 , n TP m T T n w ≤+ Δ ⋅ − 2. [] () () () 1 1 . n TP MT n W ≥+ ⋅ − Theorem 25. If is a tree, then for any T , n ∈  and n S T ⎡⎤ ⎣⎦ ∈ N 1. [] () () () () 1 , n TS m T T n w ≤+ Δ ⋅ − 2. [] () ( ) () 1 1 . n TS M T n W ≥+ ⋅ − REFERENCES [1] A.S. Asratian, R.R. Kam alian, “Interval colo rings of edges of a multigraph” , Appl. Math. 5 , pp. 25-34 , 1987. [2] A.S. Asratian, R.R. Kam alian, “Investigatio n on interval edge colorings of graphs”, J. C ombin. Theory Ser . B 62 , pp. 34-43, 1994. [3] A.S. Asratian, T.M.J. 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