L-Edge Chromatic Number Of A Graph

Transcription

1 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN L-Edge Chromatc Number Of A Graph Dr.R.B.Gnana Joth Assocate Professor of Mathematcs V.V.Vannaperuma coege for women Vrudhunagar A.Uma Dev Assocate Professor of Mathematcs V.V.Vannaperuma coege for women Vrudhunagar ABSTRACT Graph coourng s one of the most popuar concept n Graph Theory. The noton of Lst Coourng was ntroduced n 970 s by Vzng and Erdos Rubn and Tayor. In our prevous paper [] we have defned L-edge coourng of a graph. L-edge coourng of a graph s an assgnment f:e(g) X ϕ such that no two adacent edges receve the same abe where X s the ground set. A graph G s sad to be L-edge coourabe f there exsts an L-edge coourng of G. The mnmum number of prmary coours n for whch there s an L- edge coourng of G s caed the L-edge chromatc number of G and s denoted by ( ) L G.Some tmes we may use sets of specfc cardnaty as edge coours.in ths case the L-edge chromatc number s denoted as ( G ) In ths paper we determne L (...) ( G ) and L() ( G ) for some standard graphs and they are compared wth ( G) L(3) the usua edge chromatc number of G. Introducton Coour s a power n our day fe. We cannot magne a word wthout coour. Coour recreates our eyes and mnds. Due to advanced technoogy we are abe to use mutcoours. Competng brands often use ceary dfferent coours. Graph coorngs are usefu to sove varous probems rangng from schedung to the channe assgnment probem. Economcay L-edge coourng may be better than usua edge coourng. In ths paper we have found L-edge chromatc number for some standard graphs. 3

2 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN Defnton.: L-edge coourng of a graph s an assgnment L:E(G) X ϕ such that no two adacent edges receve the same abe where X s the ground set consstng of prmary coours. Defnton.: A graph G s sad to be L-edge coourabe f there exsts an L-edge coourng of G. Defnton.3: The mnmum number of prmary coours n for whch there s an L-edge coourng of G s caed the L-edge chromatc number of G and s denoted by ( ) L G. Some tmes we may use sets of specfc cardnaty as edge coours.in ths case the L-edge chromatc number s denoted as ( G ).If sngeton sets and two eement sets aone are used for edge L (...) coourng the correspondng chromatc number s denoted by ( G ). L() Note: ( G ) = ( G). L() Theorem.4: For cyces C n n 3 L() ( C n ) Proof: = = ( ) Let X = {}. Let {v v. v n } be the vertex set of C n. For a proper edge coourng we need at east two prmary coours snce ( C n ) =. Case : n = k k. In ths case the cyce contans even number of edges. Let Cvv ( ) + and ( ) Case n = k+ k C v v = k C n f s odd = for = k- f seven In ths case the cyce contans odd number of edges. - L() 33

3 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. Let Cvv ( ) + and C( v v ) k f s odd = for I = k- f s even + = {} ISSN In both the cases no two adacent edges receve the same coour. The number of prmary coours used = = ( C n ) For cyces C n ( C ) n L() = = ( ) C n Theorem.5: For path P n n ( P ) n L() = = ( ) P n Proof: Consder a path P n n. Let {v v. v n } be the vertex set and {v v v v 3. v n- v n } be the edge set of P n. For a proper coours and we can coour the edges of P n as foows: ( ) Cvv + = f s odd f s even The number of prmary coours used = - edge coourng we need ateast prmary L() For path P n ( P ) n L() = = ( ) P n Theorem.6: For a bnary tree T Proof: ( T ) L() = = ( T ) 34

4 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN v {} v u {} {} {} v u 3 v 3 u Let v be the root of the tree. Let vv and vu be the edges ncdent at v. Assgn the coours {} and to the edges vv and vu n the frst stage. In the second stage snce each vertex s of degree 3 the other two edges ncdent at v woud receve the abes and {}. Smary the two edges u u u u 3 ncdent at u woud receve the abes {} and {}. Suppose we have assgned abes for edges n (n-) th stage. At the n th stage each vertex s of degree 3. Three edges one n the (n-) th stage and two n the n th stage w be ncdent at each vertex. The edge n (n-) th stage woud have been assgned one abe from {{ } }. The remanng two abes can be assgned to other edges. Note: ( G) 3 for any graph G. Snce L() ( G) = for cyces paths and bnary L(3) trees t s not necessary to go for a L(3) -Edge coourng for them. Theorem.7: For Peterson graph G = P(n) n 4 ( G ) L() = = ( G) Proof: Let V = { } v v v u u u be the vertex set and n n 35

5 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. E={ vv vv... v v vv} 3 n n n { uu uu... u u uu} 3 n n n L() uv uv... uv n n be the edge set of P(n ). ( G) = 3 For a proper ISSN edge coourng of P(n) we need at east two prmary coours. Let X = {}. Let Cvv ( ) + ( ) C v v n f s odd = for = n- f s even = { } and Cuu ( ) Cvv ( ) f n s even f n s odd = for a. When n s even et ( ) { } When n s odd say n = k+ et Cvu ( ) Cvu = for = n for = = for = 3...k for = k + When n s even coours ncdent at v = { C(v v ) C( v v) C( uv) } When n s odd + Coours ncdent at v = { C(v v ) C( vv ) C( uv )} n = {{ } } f n s even. = {{ } } Coours ncdent at v = { C(v v) Cvv ( ) Cuv ( )} n n n n n n = {{ } } Coours ncdent at v = { C(v v ) C( v v) C( uv) } + = {{ } } 36

6 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN Any two adacent edges woud receve dfferent coours and the number of prmary coours used =. ( G ) L() = = ( G) Exampe: L() -edge coourng of P(6) v 6 v {} {} {} {} u 6 u {} {} v 5 u 5 {} {} u 4 {} u 3 u {} {} v v 4 {} v 3 L() -edge coourng of - P(5) v {} {} v 5 {} u 5 {} u {} u {} v u 4 {} u 3 {} {} {} v 4 v 3 Theorem.8: Let G = K nm be the compete bpartte graph wth n m m 4. Then ( G ) =k L() f k k < m k + k. 37

7 Proof: IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN Let V = X UY be the bpartton of k nm. Let X = n Y = m. Let X = { u u u } and Y =... n v v... v m. Snce the graph s compete bpartte every vertex n X s adacent to every vertex n Y. If there are (k-) prmary coours we have at most k k + coours for L() -edge coourng. d(u ) = m for = n To coour the edges u v = m m prmary coours are necessary. When m > k k + t s not possbe for a L() - edge coourng wth (k-) prmary coours. Let C u v = + ( ) { } = k C uv = + ( ) { } C u v = 3+ ( ) { } = 3 k = 3 4 k C uv ( ) { } + = = k For n C(u v ) = C ( uv m + + ) =... C ( uv + ) = +... m Let A = ( ) + k 38

8 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN A = ( ) k( k ) + k f + m k otherwse A 3 = ( ) k( k ) 3+ 3 k f 3+ m 3 k otherwse A k = ( ) ( ) k + = k f k + m k and s t = Note that A = { 3... m} A A = ϕ s t. Cam: C( uv ) C(u v ) for Case : Suppose A s s. Then = s + ( ) C( uv ) = s C( uv ) = s = s + ( ) for C( uv ) C(u v ) Snce they dffer n the second pace. Case : Suppose A s A t st s t Then = s + ( ) = t + ( ) ( may be equa to n ths case). C( uv ) = s 39

9 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. C( uv ) = t ISSN C( uv ) C(u v ) Snce they dffer n the frst pace. Smar proof can be gven for A A s for s and A Cam : C( uv ) C(u v ) for Wth out oss of generaty et < Case : - then - Then C(u v )=C(u v m-++ ) andc(u v )=C(u v m- ++ ) C( uv ) = C(u v ) C(u v m- ++ ) = C(u v m- ++ ) m-++=m-++ = snce < C( uv ) C(u v ) Case: - In ths case C(u v ) =C(u v -+ ) and C(u v )=C(u v m-++ ) C( uv ) = C(u v ) C(u v -+ ) = C(u v m-++ ) -+ =m-++ m- = - m=- But e between and n. So -<n m< n whch s a contradcton snce by our assumpton n m. C( uv ) C(u v ) Case 3: Snce < 40

10 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. C(u v ) =C(u v -+ ) and C(u v ) =C(u v -+ ) C( uv ) = C(u v ) + = + = snce ISSN C( uv ) C(u v ) Therefore for k nm L() = k f k k + < m k k +. That s f k k < m < k +k. Exampe: G = k 45 u u u 3 u 4 {3} {3} {} {3} {3} {} {} {} {3} {} {} {3} {} {} {3} {3} v v v 3 v 4 v 5 Theorem.9: For whee W n n 4 ( W ) n L() = k f k k < n K + k. Proof: Let v = { uv v... v v} n n be the vertex set where u s the centre vertex and E = { uv uv... uv } { v v v v... v v } be the edge set. Now coour the edges n 3 n n uv uv... uv n as foows: 4

11 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN If there are (k-) prmary coours we have at most edges. d(u) = n. k k + to coour the edges uv = n n coours are necessary. coours to coour the When n > k k + t s not possbe for a L() -edge coourng wth (k )-prmary coours. Now coour the edges uv uv uv n as foows: C uv = + ( ) C uv = + ( ) C uv = + 3 ( ) C uv = + 4 ( ) 3 = k = 3 k = 3 4 k = 4 5 k C uv ( ) { } + = = k Now we coour the edges { vv vv v v } ( ) C ( uv ) C v v n n 3 = for = n- ( ) ( ) C v v C uv n n = ( ) = C ( uv ) C v v n as foows: 4

12 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN Let A = ( ) + k A = ( ) kk ( ) + k f + n k otherwse A 3 = ( ) kk ( ) 3+ 3 k f 3+ n 3 k otherwse A k = ( ) ( ) k+ = kk + n k A = = { 3... n} and A A = ϕ m m Cam: C( uv ) C(uv ) Case : Suppose A s s Then = s + ( ) and = s + C( uv ) = s C( uv ) = s ( ) for some C( uv ) C(uv ) Snce they dffer n the second pace. Case : Suppose A s A t s 43

13 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. Then = s + = t + ( ) ( ) C( uv ) = s C( uv ) = t ISSN C( uv ) C(uv ) Snce they dffer n the frst pace. Case 3: If A then C( uv ) = { } and C( uv ) ={ } C( uv ) C(uv ) Aso for n- coours ncdent at v = { C(v v ) C( v v ) C( uv )} - + = { C(u v ) C( uv ) C( uv )} + + coours ncdent at v n = { C(v v ) C( v v ) C( uv )} n- n n n { ( n } = { C uv } C(uv n ) { } coours ncdent at v = { C(v v ) C( v v ) C( uv )} = { } n coours ncdent at v n- = { C(v v ) C( v v ) C( uv )} n- n n n n { C(u vn) C( uvn ) } = In ths coourng process no two adacent edges receve the same coour. Hence for whee W n n 4 = k ( W ) n L() f k k < n K + k. 44

14 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN Exampe: G = W 9 {} v 7 v 8 {4} {4} v 9 {4} {4} {3} {4} u {} {3} v {} {} {3} v {3} v 3 {3} v 6 {4} v 5 {3} v 4 Exampe: L() (W 6 ) = 6. A L() -edge coourng of W 6 s shown beow: v 5 {} v 6 v {} v {6} {3} v 3 v 4 {45} {35} {45} {6} {} {} v 3 {3} v 4 v {35} {5} {5} u {3} {3} {3} v 5 {5} v {5} {34} {4} {4} {4} {3} {4} {5} v 6 v 0 {5} {4} v 9 {34} {4} v 7 v 8 45

15 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN But when we go for a L(3) -edge coourng t s enough f we use fve prmary coours. Exampe: A L(3) -edge coourng of W 6 wth fve prmary coours s shown beow: v 5 {} v 6 v {} v {5} {3} {5} v 4 {5} {5} {} v 3 {3} v 3 {34} {} v 4 v {34} {34} {4} u {3} {3} {3} v 5 {34} v {4} {34} {4} {4} {4} {3} {3} {3} v 6 v 0 {34} {4} v 9 {4} {4} v 7 v 8 For the whee W 6 ( W ) = 6 whereas ( W ) L() 6 L(3) 6 = 5. For such graph L(3) - edge coourng s better than L() -edge coourng. 46

16 IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN Concuson: Generazaton of L-edge coourng for standard fames may ead to severa appcatons. In a the above resuts we note that for L and n many of them <. So our am s to check the theorems reated wth edge chromatc number L for L-edge chromatc number. References: L. B.D. Acharya Set vauatons of graphs and ther appcatons. MRI Lecture Notes n Apped Mathematcs No. MRI Aahabad 983. Dr. R.B. Gnana Joth and A.Umadev L- edge coourng of graphs presented n Natona Semnar on Recent Trends In Coourng Of Graphs And Dgraphs hed n Ramanathapuram August Courtney L. Baber Ezra Brown Char John Ross Mark Shmozono An Introducton to Lst Coorngs of Graphs Ch:4 009 Backsburg Vrgna. L 47