On the Signed Domination Number of the Cartesian Product of Two Directed Cycles

Transcription

1 Ope Joural of Dicrete Mathematic, 205, 5, Publihed Olie July 205 i SciRe O the Siged Domiatio Number of the Carteia Product of Two Directed Cycle Ramy Shahee Departmet of Mathematic, Faculty of Sciece, Tihree Uiverity, Lattakia, Syria haheeramy200@hotmailcom Received 2 March 205; accepted 2 July 205; publihed 24 July 205 Copyright 205 by author ad Scietific Reearch Publihig Ic Thi work i liceed uder the Creative Commo Attributio Iteratioal Licee (CC BY Abtract Let D be a fiite imple directed graph with vertex et V(D ad arc et A(D A fuctio :, f N v for each vertex f V ( D { } i called a iged domiatig fuctio (SDF if D [ ] V The weight ( f of f i defied by f ( v v D i { } v V The iged domiatio umber of a digraph D = mi f f i a SDF of D Let C m C deote the carteia product of directed cycle of legth m ad I thi paper, we determie the exact value of (C m C for m =, 9, 0 ad arbitrary Alo, we give the exact value of (C m C whe m, 0 (mod ad boud for otherwie Keyword Directed Graph, Directed Cycle, Carteia Product, Siged Domiatig Fuctio, Siged Domiatio Number Itroductio DV A alway mea a fiite directed graph without loop ad multiple = i the arc et If uv i a arc of D, the ay that v i a deote the et = for the A digraph D i r-regular if = The maxi-, repectively (hortly +, The δ, repectively (hortly δ +, δ Throughout thi paper, a digraph (, arc, where V = V( D i the vertex et ad A A( D + out-eighbor of u ad u i a i-eighbor of v For a vertex v V( D, let ND ( v ad ND ( v + + of out-eighbor ad i-eighbor of v, repectively We write dd( v = ND( v ad dd( v ND( v + out-degree ad i-degree of v i D, repectively (hortly d ( v, d ( v dd( v = dd( v = r for ay vertex v D Let ND[ v] = ND( v { v} ad ND[ v] ND( v { v} + mum out-degree ad i-degree of D are deoted by ( D ad ( D miimum out-degree ad i-degree of D are deoted by δ + ( D ad ( D How to cite thi paper: Shahee, R (205 O the Siged Domiatio Number of the Carteia Product of Two Directed Cycle Ope Joural of Dicrete Mathematic, 5,

2 A iged domiatig fuctio of D i defied i [] a fuctio f : V {,} uch that f N [ v] R Shahee ( D for every vertex v V The iged domiatio umber of a directed graph D i ( D = mi{ ( f f i a SDF of D} Alo, a iged k-domiatig fuctio (SKDF of D i a fuctio :, f N v k for every vertex v V The k-iged domiatio umber of a di- f V { } uch that D [ ] graph D i mi { i a SKDF of } k D = f f D Coult [2] for the otatio ad termiology which are ot defied here The Carteia product D D 2 of two digraph D ad D 2 i the digraph with vertex et V( D D2 = V( D V( D2 ad (( u, u2,( v, v2 A( D D2 if ad oly if either u = v ad ( u2, v2 A( D2 or u2 = v2 ad ( u, v A( D I the pat few year, everal type of domiatio problem i graph had bee tudied []-[7], mot of thoe belogig to the vertex domiatio I 995, Dubar et al [], had itroduced the cocept of iged domiatio umber of a udirected graph Haa ad Wexler i [], etablihed a harp lower boud o the iged domiatio umber of a geeral graph with a give miimum ad maximum degree ad alo of ome imple grid graph Zelika [] iitiated the tudy of the iged domiatio umber of digraph He tudied the iged domiatio umber of digraph for which the i-degree did ot exceed, a well a for acyclic touramet ad the circulat touramet Karami et al [9] etablihed lower ad upper boud for the iged domiatio umber of digraph Atapour et al [0] preeted ome harp lower boud o the iged k-domiatio umber of digraph Shahee ad Salim i [], were tudied the iged domiatio umber of two directed cycle C m C whe m =, 4, 5, 6, 7 ad arbitrary I thi paper, we tudy the Carteia product of two directed cycle C m ad C for m We maily determie the exact value of ( C C, ( C C, ( C C value of m ad Some previou reult: 9 ad for ome Theorem (Zelika [] Let D be a directed cycle or path with vertice The ( D = Lemma 2 (Zelika [] Let D be a directed graph with vertice The ( D ( mod 2 + Corollary (Karami et al [9] Let D be a directed of order i which d v = d v = k for each v V, where k i a oegative iteger The ( D + k I [], the followig reult are proved Theorem 4 []: C C = : 0( mod, otherwie ( C C = + 2 ( C4 C = 2 C5 C = 2 : 0( mod0, ( C5 C = 2+ :, 5, 7( mod0, C5 C = 2+ 2 : 2, 4,6,( mod0, C5 C = 2+ :, 9( mod0 ( C6 C = 2 : 0( mod, otherwie ( C6 C = 2+ 4 ( C7 C = 2 Mai Reult I thi ectio we calculate the iged domiatio umber of the Carteia product of two directed cycle C m ad C for m =, 9, 0 ad m 0( mod ad arbitrary The vertice of a directed cycle C are alway deoted by the iteger {, 2,, } coidered modulo The ith row of V ( Cm C i Ri = {( i, : =, 2,, } ad the th colum K = {( i, : i =, 2,, m} For ay vertex ( i, V ( Cm C, alway we have the idice i ad are reduced modulo m ad, repectively Let u itroduce a defiitio Suppoe that f i a iged domiatig fuctio for C m C, ad aume that h, We ay that the hth colum of f ( Cm C i a t-hift of the th colum if f ( i, = f ( i+ t, h for each vertex ( i, K, where the idice i, t, i + t are reduced modulo m ad, h are reduced modulo Remark 2: Let f i a ( Cm C -fuctio The f ( r, for each r m ad each, f i±, = f i, ± = becaue Sice C m C i 2-regular, it follow from f (( i = that ( ( f ( i,, f (( i+, = becaue f ( i+, ad f (( i, f ( i, + O the other had, if f (( i±, = f (( i, ± =, f (( i, f (( i, + =, the we mut have ((, Remark 22 Sice the cae f (( i, f (( i, 0 + = becaue + = ad f i = ice f i a miimum iged domiatig fuctio = + = i ot poible, we get 0 Furthermore, i odd if m i odd ad eve whe m i eve 55

3 R Shahee Let f be a iged domiatig fuctio for C m C, the we deote f ( K f (( i, i= colum K ad put = f ( K The equece (, 2,, i called a iged domiatig equece correpodig to f We defie The we have { } m = of the weight of a X = : = i, i = 0,,, m i X X X m = 2 f = X + 2 X + + mx m For the remaider of thi ectio, let f be a iged domiatio fuctio of C m C with iged domiatig equece (,, We eed the followig Lemma: Lemma 2 If = k the, m + 2k Furthermore, + m k ad + + m k Proof Let = k, the there are ( m k 2 of vertice i K which get value By Remark 2, K + iclude at leat 2( m k 2 of vertice which get the value ad at mot m( m k = k of vertice which ha value Hece, m + 2k Furthermore, + + m k By the ame argumet, we get m 2k ad + m k Theorem 24 : 0( mod6, + :,( mod6, ( C C = + 2 : 6,0( mod6, + : 5,7,9,( mod6, + ( + + ( + 2 C C 4 : 2, 4,,2,4 mod6, C C 5:,5 mod6, Proof We defie a iged domiatig fuctio f a follow: f (( i,2 = f (( i+ 2,2 = f (( i+ 5,2 = for 2 ad ( 6 f (( i,2 = f (( i+,2 = for 2 ad f (( i, = otherwie Alo we defie f (( i, = for i =,,, By the defiitio of f, we have = 2 for i odd ad = 4 for i eve Notice, f i a SDF for C C whe 0( mod 6,,5 mod6 Now, let u defie the followig fuctio: Therefore, there i a problem with the vertice of K whe f (( i, ( f i, if, = + if i =, 2,, 4,5,6,7,, =, ( f i, if, f (( i, = if i = 5, =, + if i =, 2,, 4,6,7,, =, We ote:, 2 ( ( f i, if, f i, = if i = 5,, =, + if i =, 2,, 4,6,7, =,, 4 ( f i a SDF of C C whe, 2, 4,,2,4,5( mod 6 f 2 i a SDF of C C whe,( mod 6 f i a SDF of C C whe 6, 9,( mod 6 f 4 i a SDF of C C whe 5,7,0 ( mod 6 Hece, ( f i, if, f i, = if i =, =, + if i =, 2,, 4,5,6,7, =, 56

4 R Shahee ( C C : 0( mod6 ( C C + :,( mod6 ( C C + 2 : 6,0( mod6 ( C C + : 5,7,9,( mod6 ( C C + 4 : 2, 4,,2,4( mod6 ( C C + 5:,5( mod6 For example, f i a SDF of C C 2, where C C2 40 = 2 + 4, ee Figure {Here, we mut ote that, for implicity of drawig the Carteia product of two directed cycle C m C, we do ot draw the arc from vertice i lat colum to vertice i firt colum ad the arc from vertice i lat row to vertice i firt row Alo for each figure of C m C, we replace it by a correpodig matrix by ig ad + which decriptio ad + o figure of f ( Cm C, repectively} By Remark 22, for ay miimum iged domiatig fuctio f of C C with iged domiatig equece (,,, we have = 0, 2, 4, 6 or for =,, By Lemma 2, if = 0 the, +, ad if = 2 the, 4 Thi implie that + ( f = Hece, by (, (2 ad ( we get for 0 mod 2 (2 = ( f = + for mod 2 ( = ( C C = for 0 mod6 ( C C = + for, mod6 Aume that / 0,,( mod6 Let f' ba a iged domiatig fuctio with iged domiatig equece (, 2,, If m, 7, the by Theorem 4 i the required (becaue Cm C C Cm Let m, We prove the followig claim: Claim 2 For k 2, we have + k d k if k i eve ad d= + + k d k whe k i odd d= + ( (a Figure (a A iged domiatig fuctio of C C 2 ; (b A correpodig matrix of a iged domiatig fuctio of C C 2 (b 57

5 R Shahee Proof of Claim 2 We have the ubequece (,, + k i icludig at leat two term The, immediately from Remark 22 ad Lemma 2, get the required The proof of Claim 2 i complete Now, if = 0 for ome, the = + = Without lo of geerality, we ca aume that 2 = 0 The Claim 2, imply that ( f = = ( = + 7 (4 = = = 4 Aume that 2 for all =,, We have three cae: Cae If = for ome Let = The from Claim 2, we get ( f = = + + ( = + 4, whe 0( mod 2 (5 Cae 2 Let 2 6 = = 2 ( f = = + + ( = + 5, whe ( mod 2 (6 = = 2 If (,, iclude at leat two term which are equal 6, the f = + 4 (7 For ( mod 2, the i eve By Lemma 2, ( C C ( f from (7 i = = = mut be eve umber Hece, f = + 5 ( Aume that 2 4 for all =,, except oce which equal 6 Thu, ( f = + 2 for 0 mod 2 (9 = ( f = + for mod 2 (0 = For the cae, we eed the followig claim: Claim 22 Let f' be a miimum iged domiatig fuctio of C C with iged domiatig equece (, 2,, The for (,,, = ( 2, 4, 2, 4, ad up to iomorphim, there i oly oe poible cofiguratio for f", it i how i Figure 2 The prove i immediately by drawig = Figure 2 The form (,,, ( 2,4,2,

6 R Shahee Cae Let 2 4 for all =,, We defie The we have X = : = i, i = 2, 4 i X2 + X4 = ( f X2 X4 = Sice the cae (, + = ( 2, 2 i ot poible, we have X4 X2 If X The If X4 = 2 + f = Thu ( f = + 4 for 0 mod 2 ( = ( f = + 5 for mod 2 (2 = f = Hece The ( f = + 2 for 0 mod 2 ( = ( f = + for mod 2 (4 = Let X4 = 2 ad X2 = 2 The we have oe poible i a the form (, 2,, = ( 2, 4, 2, 4,, 2, 4, Thi implie that ( f = for 0( mod 2 ad ( f = + for ( mod 2 By Claim 22, we have f' i a the fuctio f, which defied i forefrot of Theorem 24 However, f i ot be a iged domiatig fuctio for C C whe / 0,, mod6 Thu ( C C > for 0 mod 2 ( C C > + for mod 2 By Lemma 2, ad above argumet, we coclude that Hece, from (, (5 ad (6, deduce that Fially, we reult that: ( C C + 2 for 0 mod 2 (5 ( C C + for mod 2 (6 ( C C + 2 for 6,0 mod6 ( C C + for 5,7,9, mod6 + 2 C C + 4 for 2, 4,,2,4 mod6 + C C + 5 for,5 mod6 ( C C = for 0 mod6 ( C C = + for, mod6 ( C C = + 2 for 6,0 mod6 ( C C = + for 5,7,9, mod6 59

7 R Shahee Theorem C C + 4 for 2, 4,,2,4 mod6 + C C + 5 for,5 mod6 ( C C 9 : 0 mod, = + 6:,2 mod Proof We defie a iged domiatig fuctio f a follow: f (( i f (( i f (( i ad i ( mod 9, ad f (( i, = otherwie Alo, let u defie the followig fuctio: f (( i, if, f (( i, = + if i =, 2,, 4,5,6,7,,9, =, = +, = + 6, = for By defie f, we have = for Notice, f i a SDF for C 9 C for 0( mod of C 9 C for,2 ( mod For a illutratio (, ee Figure Hece, C C 9 6 ( C C Ad f i a SDF 9 for 0 mod (7 ( C C for, 2 mod ( From Corollary i ( C9 C The by (7, ( C9 C = for 0( mod For,2 ( mod If 4, the by Theorem 4 ad 24, get the required Aume that 9 By Remark 22, we have =,, 5, 7 or 9 By Lemma 2, if = the, + 7, = the, + ad = 5 the, + (becaue if, < +, the we eed 7 By Lemma 2, the cae (, (,, (, = are ot poible Hece, + k d d= + k, for k 2 Thi implie that, We defie The we have d = ( d (9 { } X = : = i, i =,,5,7,9 i Figure A correpodig matrix of a iged domiatig fuctio of C 9 C 6 60

8 R Shahee X+ X+ X5 + X7 + X9 = ( f X X X5 X7 X9 = If we have oe cae from the cae X 9, X 7 2, X 5 + X 7 2 or X 5 The by (9 i ( f + 6 Aume the cotrary, ie, (X 9 = 0, X 7 < 2, X 5 + X 7 < 2 ad X 5 < Hece, ( f = X+ X + 5X5 + 7X7 We coider the cae X 7 < 2 ad X 5 <, which are icludig the remaied cae, ie, X 7 = ad X 5 = 2 Firt, we give the followig Claim: Claim 2 There i oly oe poible for (, + = (, i f ( i, = f ( i+, = f (( i+ 6, = f (( i+, + = f (( i+ 4, + = f (( i+ 7, + = ad f (( i, = f (( i, + =, otherwie for i 9 The proof come immediately by the drawig Cae X 7 = ad X 5 = X 9 = 0 Without lo of geerality, we ca aume = 7 The we have the form (,,,, 7 By Claim 2, for <, each colum K + i -hift of K, K + 2 i 2-hift of K ad K + i -hift = (0-hift of K Without lo of geerality, we ca aume f ((, = f (( 4, = f (( 7, = ad f (( i, = otherwie We coider two ubcae: Subcae For ( mod The K i ( 2-hift = (2-hift of K Thi implie that f ((, = f (( 6, = f (( 9, = Hece, we eed f( ( i, = for all i =,, 9 Thi i a cotradictio with ( f ( K = 7 Thu, ( f X + 9X9 = ( + 9= + 6 Subcae 2 For 2( mod The K i ( 2-hift = (0-hift of K Thi implie that f ((, = f (( 4, = f (( 7, = So, we eed f( ( i, = for all i =,, 9 Agai, we get a cotradictio with ( f ( K = 7 Thu, ( f X + 9X9 = ( + 9= + 6 Cae 2 X 5 = 2 ad X7 = X9 = 0 Here we have k= k + d= 5 ad = otherwie By the ame argumet imilar to the Cae, we have K i ( -hift of K Thu, if ( mod, the f ((, = f (( 4, = f (( 7, = ad for 2( mod i f (( 2, = f (( 5, = f ((, = Alo, for poitio the vertice of K, we alway have f ((, = f (( 2, = f (( 4, = f (( 5, = f (( 7, = f ((, = We coider four Subcae: Subcae 2 d =, without lo of geerality, we ca aume = = 5 For ( mod, f (( 2, 2 = f (( 5, 2 = f ((, 2 = The f ((, = f (( 2, = f (( 4, = f (( 5, = f (( 7, = f ((, = The three remaiig vertice from each K ad K, mot icludig two value, ad thi i impoible The ame argumet i for 2( mod Subcae 22 d = 2, let 2 = = 5 The we have the form (, 2,, = (,,,,5,,5 If (mod, the ( mod Thi implie that K i 0-hift of K Therefore, f ((, = f (( 4, = f (( 7, = Hece, the three colum K2, K, K mut be icludig eve value of, two i K 2, three i K ad two i K ad thi impoible The ame argumet i for 2(mod Subcae 2 d =, let = = 5 We have the form (, 2,, = (,,,,5,,,5 The for ( mod, K 4 i 2-hift of K Therefore f ((, 4 = f (( 6, 4 = f (( 9, 4 = Alo, 2 = = Therefore, two vertice of {(,,( 4,,( 7, } mut ha value By ymmetry, let f (, = f (( 4, = The by Claim 2, there i oe cae for ( 2, = (, Hece, f (( 2, 2 = f (( 5, 2 = f ((, 2 = f ((, = f (( 6, = f (( 9, = Therefore, we eed two vertice from K with value Thi i a cotradictio, (becaue the vertice of the firt colum mut be a iged domiate by the vertice of the lat colum The ame argumet i for 2( mod Subcae 24 d 4, let d= = 5 (by ymmetry i d 4 We have the form (, 2,, = (,,,,5,,,,5 By Claim 2, if (, +, = (,, the for each two vertice f (( i, = f (( q, = we mut have i q = ad o for K +,, Kd Sice = ( d ad d= 5, the K d icludig two vertice with value by -hift of two vertice i K d Alo, K d + icludig two vertice with value by -hift of vertice i K d ad the third vertex mut be ditace from ay oe ha value (Sice d+ = d+ = =, Claim 2 Thu, the order of the value or of the vertice K d+,, K doe ot chage Hece the vertice of K ha the ame order of K whe we have the iged domiatig equece (,,,, ad thi impoible i iged do-,2 mod I Subcae 2, 22, 2 ad 24 there are may detail, we miatig equece of C 9 C for 6

9 R Shahee will be omitted it Fially, we deduce that doe ot exit a iged domiatig fuctio f of C 9 C for,2( mod ( f 4 ( C C + Hece, From ( ad (20 i ( C9 C 6:,2( mod Theorem 26 ( C C = with 9 + 6:,2 mod (20 = Proof We defie a iged domiatig fuctio f a follow: f (( i, = f (( i+, = f (( i+ 6, = for Alo, we defie ad ( ad i (mod 0, ad ( ( ( ( f, 7 = f 7, 7 = f 0, 7 =, ( ( ( f, 6 = f 5, 6 = f, 6 =, ( ( ( f, 5 = f 6, 5 = f 9, 5 =, ( ( ( f, 4 = f 4, 4 = f 7, 4 =, ( ( ( f 2, = f 5, = f 9, =, ( ( ( f, 2 = f 7, 2 = f 0, 2 =, ( ( ( f, = f 5, = f, =, ( ( ( f, = f 6, = f 9, =, f i, = otherwie f i, = otherwie for = 5, 4,, 2,, By defie f ad f7, f6, f5, f4, f, f2, f, f we have = 4 for all Notice that: f i a 0,, mod 0 SDF for C 0 C whe f \{ f ( K5 f ( K4 f ( K f ( K2 f ( K f ( K } { f f f f f f } { } i a SDF for C 0 C whe ( mod 0 { } { f \ f K 2 } { 2 } f K f K f f f i a SDF for C 0 C whe 2( mod 0 f \{ f ( K6 f ( K5 f ( K4 f ( K f ( K2 f ( K f ( K } { f6 f5 f4 f f2 f f} i a SDF for C 0 C whe 4( mod 0 f \ { f ( K ( 2 } f K f K f K f f 2 f f { } { } { } i a SDF for C 0 C whe 5( mod 0 { f \{ f ( K }} { f } i a SDF for C 0 C whe 6( mod 0 f \{ f ( K7 f ( K6 f ( K5 f ( K4 f ( K f ( K2 f ( K f ( K } { f7 f6 f5 f4 f f2 f f} i a SDF for C 0 C whe 7( mod 0 { } 62

10 R Shahee { f \ { f ( K 4 ( ( 2 }} { 4 2 } f K f K f K f K f f f f f i a SDF for C 0 C whe ( mod 0 { ( } { f \ f K f K } { f f } i a SDF for C 0 C whe 9( mod 0 For a illutratio ( C0 C ee Figure 4, (here for ( mod 0 the colum: K5, K4, K, K2, K, K I all the cae we have, we are chagig the fuctio of C0 C 4 By Remark 22, we have = 0, 2, 4, 6, or 0 Alo by Lemma 2, if = 0, the, + 0 ad whe = 2, i, + 6 ad = 4 i, + 4 (becaue if = 2 or + = 2, the 6 Thi implie that So, we get ( C0 C = 4 Corollary 27 For m 0( mod Proof By Corollary we have, we have C C = 4 0 = m ( Cm C = if 0( mod m 2 ( Cm C = m + m if,2( mod m ( Cm C (2 Let u a iged domiatig fuctio f a follow: f ( i2, 2 = for i m f (( i, = for i m,, ad f (( i, = for i m By defie f, we have = m/ for Notice, f i a SDF for C m C for m, 0( mod ( C C m The from (2, i ( C C = m for m, 0( mod m m For, 2(mod Let f (( i, = for i m Notice, f f ( K f Thu, ( Cm C m( m m 2m,2 mod,,, Hece, { \{ } { } } i a SDF for C m C for, 2( mod + = + Hece, by (2 i m ( Cm C m 2m + if Figure 4 A correpodig matrix of a iged domiatig fuctio of C 0 C 6

11 R Shahee Cocluio Thi paper determied that exact value of the iged domiatio umber of C m C for m =, 9, 0 ad arbitrary By uig ame techique method, our hope evetually lead to determiatio ( Cm C for geeral m ad Baed o the above (Lemma 2 ad Theorem 4, 24, 25 ad 26, alo by the techique which ued i thi paper, we agai rewritte the followig coecture (Thi coecture wa metio i []: Coecture Referece m ( C C = ( m m whe 0 mod 2 or mod [] Haa, R ad Wexler, TB (999 Boud o the Siged Domiatio Number of a Graph Dicrete Mathematic, 95, [2] Wet, DB (2000 Itroductio to Graph Theory Pretice Hall, Ic, Upper Saddle River [] Dubar, JE, Hedetiemi, ST, Heig, MA ad Slater, PJ (995 Siged Domiatio i Graph, Graph Theory, Combiatoric ad Applicatio Joh Wiley & So, Ic, Hoboke, -22 [4] Cockaye, EJ ad Myhart, CM (996 O a Geeralizatio of Siged Domiatio Fuctio of Graph Ar Combiatoria, 4, [5] Hattigh, JH ad Ugerer, E (99 The Siged ad Miu k-subdomiatio Number of Comet Dicrete Mathematic,, [6] Xu, B (200 O Siged Edge Domiatio Number of Graph Dicrete Mathematic, 29, [7] Broere, I, Hattigh, JH, Heig, MA ad McRae, AA (995 Maority Domiatio i Graph Dicrete Mathematic,, [] Zelika, B (2005 Siged Domiatio Number of Directed Graph Czecholovak Mathematical Joural, 55, [9] Karami, H, Sheikholelami, SM ad Khodkar, A (2009 Lower Boud o the Siged Domiatio Number of Directed Graph Dicrete Mathematic, 09, [0] Atapour, M, Sheikholelami, S, Haypory, R ad Volkma, L (200 The Siged k-domiatio Number of Directed Graph Cetral Europea Joural of Mathematic,, [] Shahee, R ad Salim, H (205 The Siged Domiatio Number of Carteia Product of Directed Cycle Submitted to Utilita Mathematica 64