Cyclically Interval Total Colorings of Cycles and Middle Graphs of Cycles

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1 Ope Joural of Dsree Mahemas hp://wwwsrporg/joural/ojdm ISSN Ole: ISSN Pr: Cylally Ierval Toal Colorgs of Cyles Mddle Graphs of Cyles Yogqag Zhao 1 Shju Su 2 1 Shool of See Shjazhuag Uversy Shjazhuag Cha 2 Shool of See Hebe Uversy of Tehology Taj Cha How o e hs paper: Zhao YQ Su SJ (2017) Cylally Ierval Toal Colorgs of Cyles Mddle Graphs of Cyles Ope Joural of Dsree Mahemas hps://doorg/104236/ojdm Reeved: Augus Aeped: Oober Publshed: Oober Copyrgh 2017 by auhors Sef Researh Publshg I Ths work s lesed uder he Creave Commos Arbuo Ieraoal Lese (CC BY 40) hp://reaveommosorg/leses/by/40/ Ope Aess Absra A oal olorg of a graph G s a fuo : E( G) V ( G) suh ha o adjae veres edges o de veres edges oba he same olor A k -erval s a se of k oseuve egers A ylally erval oal -olorg of a graph G s a oal olorg of G wh olors 12 suh ha a leas oe verex or edge of G s olored by = 1 2 for ay x V G S v = v e e s de o v s a { } ( ) he se { ( )} ( ) ( dg ( x ) + 1) -erval or { 12 } \ S[ x] s a ( dg ( x) 1) where d ( ) -erval G x s he degree of he verex x G I hs paper we sudy he ylally erval oal olorgs of yles mddle graphs of yles Keywords Toal Colorg Ierval Toal Colorg Cylally Ierval Toal Colorg Cyle Mddle Graph 1 Iroduo All graphs osdered hs paper are fe udreed smple graphs For a graph G le V ( G ) E( G ) deoe he se of veres edges of G respevely For a verex x V ( G) le dg ( x ) deoe he degree of x G G he maxmum degree of veres of G We deoe ( ) For a arbrary fe se A we deoe by A he umber of elemes of A The se of posve egers s deoed by A arbrary oempy subse of oseuve egers s alled a erval A erval wh he mmum eleme p he maxmum eleme q s deoed by [ pq ] We deoe [ ab ] [ ab ] he ses of eve odd egers [ ab ] respevely A erval D s alled a h -erval f D = h DOI: /ojdm O Ope Joural of Dsree Mahemas

2 Y Q Zhao S J Su suh ha o adjae veres edges o de veres edges oba he same olor The oep of oal olorg was rodued by Vzg [1] depedely by Behzad [2] The oal hroma umber χ ( G) s he smalles umber of olors eeded for oal olorg of G For a oal olorg of a graph G for ay v V ( G) le S[ v] = { ( v) } { ( e) e s de o v} A erval oal -olorg of a graph G s a oal olorg of G wh olors 12 suh ha a leas oe verex or edge of G s olored by = 1 2 for ay x V ( G) he se S[ x] s a ( dg ( x ) + 1) -erval A graph G s erval oal olorable f has a erval oal -olorg for some posve eger For ay le T deoe he se of graphs whh have a erval oal -olorg le T= T 1 For a graph G T he leas he greaes values of for whh G T are deoed by wτ ( G) Wτ ( G) respevely Clearly A oal olorg of a graph G s a fuo : E( G) V ( G) χ ( G) wτ ( G) Wτ ( G) V ( G) + E( G) for every graph G T For a graph G T le ( G) { G } θ = T The oep of erval oal olorg was frs rodued by Perosya [3] Now we geeralze he oep erval oal olorg o he ylally erval oal olorg A oal -olorg of a graph G s alled a ylally erval oal -olorg of G f for ay x V ( G) S[ x] s a ( dg ( x ) + 1) -erval or [ 1 \ S[ x] s a ( dg ( x) 1) -erval A graph G s ylally erval oal olorable f has a ylally erval oal -olorg for some posve eger For ay we deoe by F he se of graphs for whh here exss a ylally erval oal -olorg F = F 1 For ay graph G F he mmum he maxmum values of for whh G has a ylally erval oal -olorg are deoed by wτ ( G) Wτ ( G) respevely For a graph G F le Θ ( G) = { G F } I s lear ha for ay T F T F Noe ha for a arbrary graph G θ ( G) Θ ( G) I s also lear ha for ay G T he followg equaly s rue ( ) χ G wτ ( G) wτ ( G) Wτ ( G) Wτ ( G) V ( G) + E( G) A mddle graph ( ) ( ) E( G) M G of a graph G s he graph whose verex se s V G whh wo veres are adjae wheever eher hey are adjae edges of G or oe s a verex of G oher s a edge de wh I hs paper we sudy he ylally erval oal olorgs of yles mddle graphs of yles For a yle C le V ( C) = { v1 v2 v} E( C) = { e1 e2 e} where e 1 = vv 1 e = v 1v for = 23 For example he graphs Fgure 1 are C 4 M ( C 4 ) respevely Noe ha Seo 3 we always use he kd of dagram lke () Fgure 1 o deoe M C ( ) DOI: /ojdm Ope Joural of Dsree Mahemas

3 Y Q Zhao S J Su (a) (b) () Fgure 1 C M 4 ( C 4 ) (a) C (b) M 4 ( C 4 ) () Aoher dagram of M ( C 4 ) 2 C I hs seo we sudy he ylally erval oal olorgs of ( 3) ha C F ge he exa values of wτ ( C ) Wτ ( C ) se Θ ( G) I [4] was proved he followg resul Theorem 1 (H P Yap [4]) For ay eger 3 ( ) 3 f 0 mod 3 χ ( C ) 4 oherwse C show deerme he I [5] Perosya e al suded he erval oal olorgs of yles provded he followg resul Theorem 2 (P A Perosya e al [5]) For ay eger 3 we have 1) C T 2) w ( C ) τ ( ) 3 f 0 mod 3 4 oherwse 3) Wτ ( C ) = + 2 Now we osder he ylally erval oal olorgs of ( 3) o defe he oal olorg of he graph C easly we deoe V ( C) E( C) by { a a a } a v a = e for ay [ 1 ] where = Theorem 3 For ay eger 3 we have C I order 1) C F 3 f 0( mod 3 ) 2) wτ ( C ) 4 oherwse 3) Wτ ( C ) = 2 Proof Se T F he for ay G T we have χ ( G) wτ ( G) wτ ( G) So by Theorems 1 2 (1) (2) hold us prove (3) Now we show ha Wτ ( C ) 2 for ay 3 Defe a oal olorg of he graph C as follows: ( a ) [ ] = 1 2 I s easy o hek ha s a ylally erval oal 2-olorg of C Thus Wτ ( ) 2 for ay eger 3 O he oher h s easy o see ha Wτ ( C) V ( C) + E( C) = 2 So we have Wτ ( C ) = 2 Lemma 4 For ay eger 3 [ 42 2] C F 4 2 we defe a oal -olorg of he graph Proof For ay C as follows: DOI: /ojdm Ope Joural of Dsree Mahemas

4 Y Q Zhao S J Su Case 1 = 3 kk = 3 ss 2 2k 1 Subase 11 ( a ) Subase 12 3s 1 s [ 1 2k 1] ( ) ( ) ( ) f 1 ] 1 f mod 3 2 f mod3 3 f mod3 = + 4 f = 1 3 f = 2 2 f = 3 ( a ) f [ 4 [ + ] ( ) [ + ] ( ) 3 f [ + 12 ] 1( mod3 ) Subase 13 = 3s+ 2 s [ 1 2k 2] ( a ) Case 2 = 3k + 1 k = 3 ss 2 2k Subase 21 1 f mod 3 2 f 12 0 mod3 f = 1 f [ 2 1 f [ ] 0 ( mod 3 ) 2 f [ + 12 ] 1( mod3 ) 3 f [ + 12 ] 2( mod3 ) 4 f = 1 3 f = 2 2 f = 3 Subase 22 3s 1 s [ 1 2k 1] ( a ) f [ 4 1 f [ ] 1 ( mod 3 ) 2 f [ + 12 ] 2( mod3 ) 3 f [ + 12 ] 3( mod3 ) = + ( a ) 4 f = 1 f [ 2 1 f [ ] 2 ( mod 3 ) 2 f [ + 12 ] 0( mod3 ) 3 f [ + 12 ] 1( mod3 ) DOI: /ojdm Ope Joural of Dsree Mahemas

5 Y Q Zhao S J Su Subase 23 3s 2 s [ 12k 1] = + ( a ) Case 3 = 3k + 2 k = 3 ss 2 2k Subase 31 ( ) ( ) ( ) f 1 1 f mod 3 2 f mod3 3 f mod3 DOI: /ojdm Ope Joural of Dsree Mahemas ( a ) Subase 32 3s 1 s [ 1 2k] f = 1 f [ 2 1 f [ ] 1 ( mod 3 ) 2 f [ + 12 ] 2( mod3 ) 3 f [ + 12 ] 3( mod3 ) = + ( ) ( ) ( ) f 1 1 f mod 3 ( a ) 2 f mod3 3 f mod3 Subase 33 3s 2 s [ 1 2k] = + 4 f = 1 3 f = 2 2 f = 3 ( a ) f [ 4 1 f [ ] 0 ( mod 3 ) 2 f [ + 12 ] 1( mod3 ) 3 f [ + 12 ] 2( mod3 ) I s o dfful o hek ha eah ase s always a ylally erval oal -olorg of C The proof s omplee Lemma 5 C3 F 5 Proof We defe a oal 5-olorg of he graph C 3 as follows: ( a ) f f = 6 I s easy o see ha s a ylally erval oal olorg of C 3 Lemma 6 For ay eger 4 C F 2 1 Proof By orado Suppose ha for ay egers 4 s a ylally erval oal ( 2 1) -olorg of suh ha ( a) ( j) C The here exs dffere j [ 12] = a for dffere s [ 1 2 ] \ { j} a a ( ) ( j)

6 Y Q Zhao S J Su a Whou loss of geeraly we may assume ha ( ) ( ) 1 [ 22 1] = a = The for eah k here s oly oe verex or oe edge of C s olored by k Case 1 A leas oe of j s eve Say ha s eve Whou loss of geeraly suppose ha = 2 e ( a2 ) = 1 The we have 3 j 2 3 Noe ha a2 = vv 1 Se s a ylally erval oal ( 2 1) -olorg of C he we have { ( v1) ( vv 1 2) } = { 23 }{ 2 2 1} or { } j Beause { ( v) ( v 1v) } = { } { } { } or { } { ( v1) ( vv 1 2) } ( v) ( v 1v) = whou loss of geeraly we may assume ha Se ha for eah k [ 22 1] { ( v1) ( vv 1 2) } = { 23} { ( v) ( v 1v) } = { } here s oly oe verex or oe edge of C s olored by k The ( aj 1) [ 4 2 3] or ( aj+ 1) [ 42 3] O he oher h se s a ylally erval oal ( 2 1) -olorg of C he ( aj 1) ( aj+ 1) { } weaher j s odd or eve A orado Case 2 j are all odd Whou loss of geeraly suppose ha = 1 The we have 3 j 1 Noe ha a a j are all veres of C Se s a ylally erval oal ( 2 1) -olorg of C he we have { ( a2) ( a2 )} = { 23 }{ 2 2 1} or { } Beause { ( aj 1) ( aj+ 1) } = { } { } { } or { } { ( a2) ( a2) } ( aj 1) ( aj 1) + = whou loss of geeraly we may assume ha { ( a2) ( a2 )} = { 23} { ( aj 1) ( aj+ 1) } = { } say ( a 2 ) = 2 The ( a2 ) 3 defo of ( a3 ) { } Bu ( a 3 ) obvously So we have ( a 3 ) = 4 he ( ) = Now we osder he olor of a 3 By he a o be 1 3 or 2 1 a 4 = 3 Ths s a orado o DOI: /ojdm Ope Joural of Dsree Mahemas

7 Y Q Zhao S J Su ha jus oe verex or oe edge of Se we already have ( a ) = before 2 3 C s olored by where [ 22 1] Combg Theorem 3 Corollares 4-6 he followg resul holds Theorem 7 For ay eger 3 3 M( C ) [ 3 2 ] f 3 { } ( ) [ 4 2 ] \ { 2 1 } oherwse = Θ ( C ) 3 2 \ 2 1 f 4 0 mod 3 I hs seo we sudy he ylally erval oal olorgs of ( ) ( 3) prove M ( C ) F ge he exa values of wτ ( M ( C )) of Wτ ( M ( C )) show ha for ay k bewee w ( M ( C )) boud of Wτ ( M ( C )) M ( C) F k Theorem 8 For ay eger 3 ( ) Proof Suppose ha eger 3 he graph M ( C ) as follows: Case 1 s eve ( ) 5 M C provde a lower boud τ he lower wτ M C = Now we defe a oal 5-olorg of ( v ) [ ] = 3 1 ( e ) [ ] [ ] ( ev ) [ ] = 2 1 ( ve ) + 1 ( ee ) + 1 where e+ 1 = e1 See Fgure 2 By he defo of we have [ ] [ ] [ ] [ ] [ ] = [ 13 ] [ 1 ] [ ] = [ 24 ] [ 1 ] S v S v Fgure 2 A oal 5-olorg of M ( C 4 ) DOI: /ojdm Ope Joural of Dsree Mahemas

8 Y Q Zhao S J Su [ ] = S e 15 1 Case 2 s odd ( v ) [ ] = [ ] ( e ) = ( ve ) + 1 ( ev ) [ ] = [ ] { } [ ] ( ve 1 ) 5 3 [ 1 1 ] 5 [ 1 1 ] ( ee ) + 1 = ( ) ee 1 = 4 See Fgure 3 By he defo of we have S v = [ ] { } S[ v ] = [ 24 ] [ 1 2 ] [ ] = [ 35 ] S v [ ] = S e 15 1 Combg Cases 1 2 we kow ha for ay eger 3 s a ylally erval oal 5-olorg of M ( C ) Therefore ( ) ( ) 5 w M C τ Fgure 3 A oal 5-olorg of M ( C 5 ) DOI: /ojdm Ope Joural of Dsree Mahemas

9 Y Q Zhao S J Su O he oher h So we have ( ( ) ) ( ( ) ) 1 5 w M C M C τ + = ( ( )) 5 wτ M C = Theorem 9 For ay eger 3 Wτ ( M ( C )) 4 Proof Now we defe a oal 4-olorg of he graph ( ) where [ 1 ] ( v ) 4 1 = ( e ) 4 3 = ( ev ) 4 2 = ( ve ) + 1 = 4 ( ee ) + 1 = 4 1 e+ 1 = e1 See Fgure 4 By he defo of we have [ ] = S v [ ] = [ ] S e [ ] = S e M C as follows: Ths shows ha s a ylally erval oal 4-olorg of M ( C ) So we have Wτ ( M ( C )) 4 Theorem 10 For ay eger 3 ay k [ 54] M ( C) F k Proof Suppose 3 for ay k [ 54] We defe a oal k -olorg of M ( C ) as follows Frs we use he olors 1 2 k o olor he veres edges of ( ) M C begg from e 1 by he way used he proof M C wh he of Theorem 9 Now we olor he oher veres edges of ( ) olors 1 2 k Fgure 4 A oal 16-olorg of M ( C 4 ) DOI: /ojdm Ope Joural of Dsree Mahemas

10 Y Q Zhao S J Su k k = The we have ( ve + 1) = k where 1 4 Subase 11 s eve v + = 3 1 Case 1 0( mod 4) ( ) 1 [ 1 ( e + ) 4 [ 1 ( e + v + ) = 2 [ 1 1 [ 1 ( v e 1) 4 [ [ 1 ( e + e ++ 1) 5 [ 1 where e+ 1 = e1 See Fgure 5 By he defo of we have S v = [ ] [ + ] = [ 13 ] [ 1 ] [ + ] = [ 24 ] [ 1 ] S[ e1 ] = [ 15 ] [ ] = [ ] S[ e + ] = [ 13] [ k 1 k] S v S v S e [ + ] = S e 15 2 Subase 12 s odd v + = ( ) ( v ) = ( e + ) 4 1 Fgure 5 A oal 8-olorg of M ( C 6 ) DOI: /ojdm Ope Joural of Dsree Mahemas

11 Y Q Zhao S J Su reolor e Fgure 6 ( e v ) [ ] + + = By he defo of we have k k Subase 21 s eve Case 2 1( mod 4) ( ev ) 3 = 1 1 ( v + e ++ 1) 4 1 [ ] ( e + e ++ 1) ( ee) 1 = 2 ev as ( e 1 ) = 4 ( ev ) S[ v1 ] = [ 35 ] S[ v ] = [ 4 24 ] [ 2 S[ v + ] = [ 13 ] [ 1 S[ v + ] = [ 24 ] [ 1 S[ e1 ] = [ 15 ] [ ] = [ ] S[ e + ] = [ 13] [ k 1 k] S e [ + ] = S e 15 2 = The we have ( e ) = k where 2 ( v ) = k ( v+ ) = 3 [ 1 1 [ 1 ( e ) 4 [ = 5 where e 1 e1 + = See Fgure 6 A oal 8-olorg of M ( C 5 ) DOI: /ojdm Ope Joural of Dsree Mahemas

12 Y Q Zhao S J Su ( ev ) = 1 ( e + v + ) = 2 [ 1 ( ve + 1 ) = k 1 1 [ 1 ( v e 1) 4 [ 1 ( ee + 1 ) = k 3 [ 1 ( e e 1) 5 [ reolor e = where e+ 1 = e1 See Fgure 7 By he defo of we have S v = e as ( ) 2 Subase 22 s odd [ ] S[ v ] = { 1 k 1 k} S[ v + ] = [ 13 ] [ 1 S[ v + ] = [ 24 ] [ 1 S[ e1 ] = [ 1 5 ] [ ] = [ ] S[ e ] = [ 12] [ k 2 k] S[ e + ] = [ 13] [ k 1 k] S e [ + ] = S e 15 2 ( v ) = 1 ( v ) 3 [ 1 4 [ 1 1 [ 1 + = ( e + ) e + v + = 2 0 ( ) Fgure 7 A oal 9-olorg of M ( C 5 ) DOI: /ojdm Ope Joural of Dsree Mahemas

13 Y Q Zhao S J Su where e+ 1 = e1 See Fgure 8 By he defo of we have k k Subase 31 s eve Case 3 2( mod 4) ( ve ) + 1 = ( v + e ++ 1) 1 1 ( ee ) + 1 = ( e + e ++ 1) 3 1 S[ v ] = [ 4 24 ] [ 1 1 ] S[ v + ] = [ 24 ] [ 0 S[ v + ] = [ 13 ] [ 0 S[ e1 ] = [ 1 5 ] [ ] = [ ] S[ e ] = [ 12] [ k 2 k] S e [ + ] = S e 15 1 = The we have ( ev ) = k where 2 ( v ) = k ( v+ ) = 3 [ 1 1 [ 1 ( e ) 4 [ 1 ( e v ) 2 [ 1 ( ve+ ) = k = ( v + e ++ 1) 4 1 Fgure 8 A oal 9-olorg of M ( C 6 ) DOI: /ojdm Ope Joural of Dsree Mahemas

14 Y Q Zhao S J Su ( ee ) + = 1 k 3 1 ( e + e ++ 1) 5 1 reolor ev as ( ev ) = 1 where e+ 1 = e1 See Fgure 9 By he defo of we have S v = Subase 32 s odd [ ] S[ v ] = { 1 k 1 k} S[ v + ] = [ 13 ] [ 1 S[ v + ] = [ 24 ] [ 1 S[ e1 ] = [ 1 5 ] [ ] = [ ] S[ e ] = { 1} [ k 3 k] S[ e + ] = [ 13] [ k 1 k] S e [ + ] = S e 15 2 ( v ) = 1 ( v + 1 ) = 2 ( v+ ) = 3 [ 2 4 [ 1 ( e + ) 1 [ 1 ( e + 1v + 1) = 3 ( e + v + ) = 2 [ 2 ( ve + 1 ) = 2 4 [ 1 ( v e 1) 1 [ Fgure 9 A oal 10-olorg of M ( C 5 ) DOI: /ojdm Ope Joural of Dsree Mahemas

15 Y Q Zhao S J Su where e+ 1 = e1 See Fgure 10 By he defo of we have k k Subase 41 s eve Case 4 3( mod 4) reolor 1 ( ee ) + 1 = ( e + e ++ 1) 3 1 S[ v ] = [ 4 24 ] [ 1 1 ] S[ v ] = { 1 2 k} S[ v + ] = [ 24 ] [ 1 S[ v + ] = [ 13 ] [ 1 S[ e1 ] = [ 1 5 ] [ ] = [ ] S e [ ] = { } S e 1 k 3 k [ + ] = S e 15 1 = The we have ( v ) e as ( ) = k where 2 ( v+ ) = 3 [ 1 4 [ 1 ( e + ) 1 [ 1 ( e + v + ) = 2 [ 1 4 [ 0 ( v e 1) 1 [ ( e + e ++ 1) 5 1 e 1 = 4 where e+ 1 = e1 See Fgure 11 Fgure 10 A oal 10-olorg of M ( C 6 ) DOI: /ojdm Ope Joural of Dsree Mahemas

16 Y Q Zhao S J Su Fgure 11 A oal 7-olorg of M ( C 6 ) By he defo of we have Subase 42 s odd where e+ 1 = e1 See Fgure 12 [ ] = S[ v ] = { 1 k 1 k} S[ v + ] = [ 24 ] [ 1 S[ v + ] = [ 13 ] [ 1 S[ e1 ] = [ 1 5 ] [ ] = [ ] S[ e + 1] = [ 14 ] { k} S[ e + ] = [ 15 ] [ 2 S v S e ( ) + 1 = 4 v ( v ) [ + = 3 2 ( ) e + 1 = ( e + ) 1 2 ( e v ) + + = ( e v ) [ + + = 2 2 ( ve ) + 1 = 1 ( v e ) + + = ( v + e ++ 1) 1 2 ( e e ) + + = ( e + e ++ 1) 3 2 DOI: /ojdm Ope Joural of Dsree Mahemas

17 Y Q Zhao S J Su Fgure 12 A oal 7-olorg of M ( C 5 ) By he defo of we have [ ] = S[ v ] = { 1 k 1 k} S[ v + 1] = [ 35 ] S[ v + ] = [ 24 ] [ 2 S[ v + ] = [ 13 ] [ 2 S[ e1 ] = [ 15 ] [ ] = [ ] S[ e + ] = [ 14 ] { k} S v S e [ + ] = S e 15 2 Combg Cases 1-4 he resul follows 4 Coludg Remarks I hs paper we sudy he ylally erval oal olorgs of yles mddle graphs of yles For ay eger 3 we show 0 mod 3 ) or 4 (oherwse) W ( C ) 2 as ( ) C F prove ha wτ ( C ) = 3 (f τ = deerme he se Θ ( G) [ 3 2 ] f = 3 Θ ( C ) [ 3 2 ] \ { 2 1 } f 4 0( mod 3 ) [ 4 2 ] \ { 2 1 } oherwse we have M ( C ) F prove ha ( ( )) 5 ( ) ojeure ha ( ) For ay eger 3 wτ M C = Wτ ( M C ) 4 for ay k bewee 5 4 M ( C) F k We Wτ ( M C ) = 4 I would be eresg fuure o sudy he ylally erval oal olorgs of graphs relaed o yles Akowledgemes We hak he edor he referee for her valuable ommes The work was DOI: /ojdm Ope Joural of Dsree Mahemas

18 Y Q Zhao S J Su suppored par by he Naural See Foudao of Hebe Prove of Cha uder Gra A par by he Isue of Appled Mahemas of Shjazhuag Uversy Referees [1] Vzg VG (1965) Chroma Idex of Mulgraphs Dooral Thess Novosbrsk ( Russa) [2] Behzad M (1965) Graphs Ther Chroma Numbers PhD Thess Mhga Sae Uversy Eas Lasg MI [3] Perosya PA (2007) Ierval Toal Colorgs of Complee Bpare Graphs Proeedgs of he CSIT Coferee Yereva [4] Yap HP (1996) Toal Colorgs of Graphs ure Noes Mahemas 1623 Sprger-Verlag Berl [5] Perosya PA Torosya AYu Khaharya NA (2010) Ierval Toal Colorgs of Graphs arxv: v1 DOI: /ojdm Ope Joural of Dsree Mahemas