Lower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori

Lower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori Petros A. Petrosyan Institute for Informatics and Automation Problems of NAS of RA, Department of Informatics and Applied Mathematics, Y SU, Yerevan, Arm enia e-mail: pet _petros@ipi a.sci.am, pet_petros@ya hoo.com Gagik H. Karapetyan Department of Informatics and Applied Mathematics, Y SU, Yerevan, Arm enia e-mail: kar_gagik@yahoo.com ABSTRACT An interval edge t − coloring of a graph G is a proper edge coloring of G with colors 1, 2 , , t … such that at least one edge of G is colored by color , 1 ,2 , , ii t = … , and the edges incident with each vertex () vV G ∈ are colored by ( ) G dv consecutive colo rs, where ( ) G dv is the degree of the vertex v in G . In this paper interval edge colori ngs of bipartite cylinders and bipart ite tori are investigated. Keywords Interval edge coloring, prop er edge coloring , bipartite graph . 1. INTRODUCTION All graphs considered in this paper are finite, undirected and have no loops or multiple edges. L et () VG and () EG denote the sets of vertices and edges of a grap h G , respectively. Th e degree of a vertex () vV G ∈ is denoted by ( ) G dv , the maximum degree of a vertex of G - by () G ∆ , the chromatic index of G - by () G χ ′ , and the diameter of G - by () dG . Given two graphs ( ) 11 1 , VE G = and ( ) 22 2 , VE G = , the Cartesian product 12 GG × is a graph () , VE G = w ith vertex set 12 VV V =× and the edge set () ( ) () () { 12 1 2 1 1 22 2 2 2 ,, , , either and or and uu vv u v u v E u v E =∈ = = () } 11 1 , uv E ∈ . The bipartite cylinder () ,2 mn C is the Cartesian product () 2 , 2 mn Cm N n P ×∈ ≥ and the bipartite torus ( ) 2, 2 mn T is the Cartesian product () 22 2, 2 mn Cm n C ×≥ ≥ . If α is a proper edge coloring of the graph G then () e α denotes the color of an edge () eE G ∈ in the coloring α . For a proper edge coloring α of a graph G and for any () vV G ∈ we denote by (, ) Sv α the set of colors of edges incident with v . An interval [1] edge t − coloring of a graph G is a proper edge coloring of G with colors 1, 2 , , t … such that at least one edge of G is colored by color , 1 ,2 , , ii t = … , and the edges incident with each vertex () vV G ∈ are colo red by ( ) G dv consecutive colors. For 1 t ≥ let t N denote the set of graphs which have an interval edge t − coloring, and assume: 1 t t ≥ ≡ ∪ N N . For a graph G ∈ N the least and the greatest values of t , for which t G ∈ N , are denoted by () wG and () WG , respectively. The problem of decid ing whether or not a bipartite graph belongs to N was shown in [2] to be NP − complete [3,4 ]. It was proved in [5] that if ( ) ,2 mn GC = or ( ) 2, 2 mn GT = then G ∈ N and () () wG G =∆ . Theorem 1 [6]. If G is a bipartite graph and G ∈ N then ( ) () 1 1 () () G WG d G ∆ −+ ≤ . Theorem 2 [7]. Let G be a regular graph. 1. G ∈ N iff () ( ) GG χ ′ = ∆ . 2. If G ∈ N and () () Gt W G ∆ ≤≤ then t G ∈ N . In this paper interval edge colorings of bipartite c ylinders and bipartite tori are investi gated. The terms and concepts that we do not define can be found in [8-10] . 2. LOWER BOUNDS FOR ( ) ( ) ,2 WC m n AND ( ) ( ) 2, 2 WT m n . Theorem 3. If ( ) ,2 mn GC = then () 3 2 WG m n ≥+ − . Proof. Let { } () 2 11 1, 1 2 , () () () , () i j mn i j ij xi m j n EG E G E G VG == ≤≤ ≤ ≤   =     = ∪ ∪∪ where ( ) { } ( ) { } () () () () 11 2 , 1 2 1 , , () i ii ii jj n xx j n xx EG + ≤≤ − = ∪ () { } () ( 1 ) , 1 1 . () ii jj j xx i m EG + ≤≤ − = Define an edge coloring α of the graph G in the following way: 1. for 1, 2 , , , 1, 2 , , 1 im jn = =+ …… ( ) ( ) () () 1 ,3 3 ii jj x xi j α + =+ − ; 2. for 1 , 2, , , 2, , 2 1 im j n n = =+ − …… ( ) ( ) () () 1 ,3 2 1 ii jj xx i j n α + =− + − ; 3. for 1 , 2, , im = … ( ) ( ) () () 12 ,3 1 ii n x xi α =− ; 4. for 1, 2 , , 1, 2 , 3 , , 1 im j n =− = + …… ( ) ( ) () ( 1 ) ,3 2 ii jj xx i j α + =+ − ; 5. for 1, 2 , , 1, 2 , , 2 im j n n =− = + …… ( ) ( ) () ( 1 ) ,3 2 1 ii jj xx i j n α + =− + + ; 6. for 1, 2 , , 1 im =− … ( ) ( ) () ( 1 ) 11 ,3 ii x xi α + = . Let us show that α is an interval edge ( ) 32 mn + −− − coloring of the graph G . First of all let us prove that for , 1 ,2 , ii = … ,3 2 mn +− … there is an edge () i EG e ∈ such that () i i e α = . For 1, 2 , , im = … we define a set i F in the following way: () ( ) { } () () 1 , 1 1 ii jj i xx j n F α + ≤≤+ = . Clearly, {} 3 2,3 1 , ,3 2 , 1 , 1 , 2, , i i ii i n F n i m F −− + − = + = = …… . It is not hard to check that {} 1 1, 2 , , 3 2 m i i Fm n = =+ − … ∪ , and, therefore for , 1 ,2 , , 3 2 ii m n =+ − … there is an edge () i EG e ∈ such that () i i e α = . Now, let us sho w that the edge s that are incid ent to a vertex () vV G ∈ are colored by ( ) G dv consecutive colors . Let () () i j x VG ∈ , where 1, 1 2 im j n ≤≤ ≤ ≤ . Case 1. 1, 1, 2 ij == . It is not hard to see that ( ) {} () ,3 2 , 3 1 , 3 i j Sx i i i α =− − . Case 2. 1, 3 , , 2 ij n == … . It is not hard to see that ( ) ( ) { } () () 23 3, , 1 . ,, 1 , , 1 , where ii kn k kn Sx Sx k k k αα +− =+ == − + … Case 3. ,1 , 2 im j == . It is not hard to s ee that ( ) {} () ,3 3 , 3 2 , 3 1 i j Sx i i i α =− − − . Case 4. ,3 , , 2 im j n == … . It is not hard to s ee that ( ) ( ) { } () () 23 3, , 1 . , , 35 , 3 4 , 33 , where ii kn k kn Sx Sx ik ik ik αα +− =+ = = +− +− +− … Case 5. 2, 3 , , 1 , 1 , 2 im j =− = … . It is not hard to s ee that ( ) {} () ,3 3 , 3 2 , 3 1 , 3 i j Sx i i i i α =− − − . Case 6. 2, 3 , , 1 , 3, , 2 im j n =− = …… . It is not hard to s ee that ( ) ( ) { } () () 23 , 2 , , 2 . , , 32 2 , 32 1 , 32 , 32 1 w h e r e ii kn k kn n Sx Sx i k n ik n ik n ik n αα +− =+ == − + − − + − −+ −+ + … Therefore, α is an interval edge ( ) 32 mn +− − coloring of the graph G . The proof is com plete. Remark. Since ( ) ,2 mn C is a bipartite graph with () ( ) ,2 24 Cm n ≤ ∆≤ and () ( ) ,2 1 Cm n m n d =+ − then from theorem 1 we have () ( ) ,2 33 2 Cm n Wm n ≤+ − . Theorem 4. If ( ) 2, 2 mn GT = then { } 3, 3 () m a x mn nm WG ++ ≥ Proof. Let { } () 12 , 1 2 , and ( ) i j x im j n mn V G ≤≤ ≤ ≤ ≤= 22 11 () () () , mn i j ij EG E G E G ==   =     ∪ ∪∪ where ( ) { } ( ) { } () () () () 11 2 , 1 2 1 , , () i ii ii jj n xx j n xx EG + ≤≤ − = ∪ ( ) { } () { } () ( 1 ) ( 1 ) ( 2 ) , 1 2 1 , . () ii m jj jj j xx i m xx EG + ≤≤ − = ∪ Define an edge coloring β of the graph G in the following way: 1. for 1, 2 , , , 1, 2 , , 1 im jn = =+ …… ( ) ( ) ( ) ( ) ( ) ( ) (2 1 ) (2 1 ) 11 ,, 3 3 i i mi mi jj j j x xx x i j ββ +− +− ++ = =+ − ; 2. for 1 , 2, , , 2, , 2 1 im j n n = =+ − …… ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1) ( 2 1) 11 ,, 3 6 3 ii m i m i jj j j xx x x i j n ββ +− +− ++ = =− + + ; 3. for 1, 2 , , im = … ( ) ( ) ( ) ( ) ( ) ( ) (2 1 ) (2 1 ) 12 1 2 ,, 3 i i mi mi nn x xx x i ββ +− +− = =+ ; 4. for 1 , 2, , , 2, 3 , , 1 im j n = =+ …… ( ) ( ) ( ) ( ) () ( 1 ) ( 2 ) ( 2 1 ) ,, 3 4 ii m i m i jj j j x xx x i j ββ +− + − = =+ − ; 5. for 1 , 2, , , 2, , 2 im j n n = =+ …… ( ) ( ) ( ) ( ) () ( 1 ) ( 2 ) ( 2 1 ) ,, 3 6 5 ii m i m i jj j j xx x x i j n ββ +− + − = =− + + ; 6. for 1, 2 , , im = … ( ) ( ) ( ) ( ) () ( 1 ) ( 2 ) ( 2 1 ) 11 1 1 ,, 2 ii m i m i x xx x i ββ +− + − = =+ ; 7. for 3, , 1 jn = + … ( ) ( ) ( ) ( ) (1) ( 2 ) (1) ( 2 ) 23 23 ,, 3 4 mm j j nj nj xx x x j ββ +− +− = =− ; 8. ( ) ( ) ( ) ( ) ( 1 ) ( 2) ( 1 ) ( 2) 11 22 ,, 2 mm xx xx ββ == . Let us show that β is an interval edge ( ) 3 nm + − − coloring of the graph G . First of all let us prove that for , 1 ,2 , ii = … ,3 nm + … there is an edge () i EG e ∈ such that () i i e β = . It is not hard to c heck that () {} 1 () 1 1 , 1, 2 , , 3 m n i j i j Sx nm β + = = =+ ∪ … ∪ , and, therefore for , 1 ,2 , , 3 ii n m = + … there is an edge () i EG e ∈ such that () i i e β = . Now, let us sho w that the edge s that are incid ent to a vertex () vV G ∈ are colored by four consecutive colors. Let () () i j x VG ∈ , where 12 , 1 2 im jn ≤≤ ≤ ≤ . Case 1. 1, 2 , , 2 , 1 im j = = … . It is not hard to s ee that ( ) ( ) {} () ( 2 1 ) 1, 2 , , . ,, , 1 , 2 , 3 , where km k jj km Sx Sx k k k k ββ +− = = =+ + + … Case 2. 1, 2 , , 2 , 2 , 3 , , 1 im j n == + …… . It is not hard to see that () ( ) { } () ( 2 1 ) , 1, 2 , , . ,, 3 6 , 3 5 , 34 ,33 w h e r e km k jj km Sx Sx k j k j kj kj ββ +− = == + − + − +− +− … Case 3. 1, 2 , , 2 , 2 , , 2 1 im j n n == + − …… . It is not hard to see that () ( ) { } () ( 2 1 ) 1, 2 , , . ,, 3 6 3 , 3 6 4 , 365 ,36 6 , w h e r e km k jj km Sx Sx k j n k j n kj nkj n ββ +− = == − + + − + + −+ + −+ + … Case 4. 1, 2 , , 2 , 2 im j n == … . It is not hard to see that ( ) ( ) {} () ( 2 1 ) 1, 2 , , . ,, 3 , 4 , 5 , 6 , where km k jj km Sx Sx k k k k ββ +− = == + + + + … This shows that β is an interval edge () 3 nm +− coloring of the graph G . The proof is com plete. From theorem 2 and theo rem 4 we have the fo llowing Corollary. If ( ) { } 2, 2 , 4 m a x 3 , 3 GT m n t m n n m =≤ ≤ + + then t G ∈ N . REFERENCES [1] A.S. Asratian, R.R. Kamalian , “Interval colorings of edges of a multigraph”, Appl. Math . 5 , pp. 25-34 , 1987. [2] S.V. 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