BULLETIN of the Bull. Malaysa Math. Sc. Soc. (Secod Seres) (00) 9-0 MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Uquely N-colorable Chromatcally Equvalet Graphs CHONG-YUN CHAO Departmet of Mathematcs, Uversty of Pttsburgh, Pttsburgh, PA 50 Abstract. For each teger, we preset a uquely -colorable graph wth vertces, ther geeralzatos. For each teger, we preset two uquely ( ) - colorable graphs whch are chromatcally equvalet wth vertces, wth vertces, ther geeralzatos.. Itroducto Let G be a smple graph, V(G), be ts vertex-set E(G) be ts edge-set. A assgmet of colors to the vertces of G such a way that adacet vertces are assged wth dfferet colors s called a (proper) colorg of G. The mmum umber of colors eeded to color G, s called the chromatc umber of G, s deoted by (G). Let be a postve teger. The a -colorg of G s a partto of V (G) to color classes such that the vertces the same color class are ot adacet. If every (G) -colorg of G gves the same partto of G, the G s sad to be a uque (G) -colorable graph. A chromatc polyomal, P ( G, ), s the umber of ways of -colorg of G. Two graphs G H are sad to be chromatcally equvalet, f they are osomorphc P( G, ) P( H, ). I [], some famles of uquely -colorable graphs wthout tragles were preseted. I [], the author, Osterwel, preseted some famles of uquely -colorable graphs wth tragles. Hs method was to use the complemets of certa graphs. He also stated that the techques used here seem applcable to the more geeral study of uque -colorablty graphs. Recetly, Cha [], by usg the same method as Osterwel's, exteded Osterwel's result to the case of uquely -colorable graphs. Here, we shall also cosder the complemets of certa graphs to prove the followg Theorem. (a) For each teger, there exsts a uquely -colorable graph wth vertces. (b) For each teger, there exsts a uquely -colorable graph wth vertces.
Usg our Theorem, we shall prove Theorem. (a) For each teger, there exst two uquely (+)-colorable graphs wth vertces whch are chromatcally equvalet. (b) For each teger, there exst two uquely (+)-colorable graphs wth vertces whch are chromatcally equvalet. We eed the followg well kow Theorem (see p. 55 []) for our proofs examples: Let G be a graph. The P( G, ) P( G e, ) P( G / e, ) () where G e s the graph obtaed from G by deletg a edge e G, G / e s the graph obtaed from G by cotractg the edge e. Or P( G, ) P( G e, ) P(( G e) / e, ) () where e E (G) G e s the graph obtaed from G by addg the edge e to G.. Examples, proofs geeralzatos The followg examples lead to the proof for the geeral case,.e., they lead to the proofs for Theorem (a) (b). Example. Let P 5 be the followg path of legth 5, P 5 0 5 be the complemet of P the complete graph,, wth vertces. Thus, G 5 s the followg graph: K 0 G 5
Sce cotas the tragles K 0,, K,, 5, ( ). We clam that G ca be colored by colors,, wth color dfferece. We shall use the followg otatos: ( ) meas the vertex s colored by the color, () meas the vertex s forced to be colored wth the color. We color ( ), ( ) 5( ). Sce the eghborhood of the vertex 0, N (0) {( ), 5( )}, 0( ). Smlarly, sce N( ) {0( ), 5( )}, ( ), sce N( ) {( ), ( ), 0( )}, ( ). Thus, ( ). Sce there s o choce of colors for each vertex, s uquely -colorable, the color classes are {0, }, {,}, {,5}. We also ca show that G s a uquely -colorable graph by usg ts chromatc polyomal. The chromatc polyomal of G s, by addg e ( 0, ) to G usg () deletg e (, ) e usg (), P(, ) ( P( e, ) P( / e, ) P(( G e ) e, ) e, ) P( e ) / e, ) P( / e, ( )( ) ( ) ( )( )( ) ( )( ) The Lemma [] states: Let ( G) k. The P( G, k) k! t for some postve teger t, t s the umber of ways of colorg G exactly k colors wth color dfferece. Furthermore, t f oly f G s a uquely k-colorable graph. Hece, wth, P( G, ) 0 0!, G s a uquely -colorable graph. Example. Let H be the graph wth vertces cosstg of a -cycle a tragle,.e., V ( H ) {0,,, } E( H ) {(0,), (, ), (, ), (, ), (, 5), (5, ), (, 0), (,)}, H be the complemet of H K. Thus, H s the followg graph: ). 0 H 5
Sce H cotas tragles, ( H ). We clam that H ca be colored by colors, wth color dfferece. We color ( ), ( ) 5( ). The sce N( 0) {( ), 5( )}, 0( ). Smlarly, sce N ( ) {0( ), 5( )}, ( ). Sce N( ) {0( ), ( ), ( )}, ( ). Sce N ( ) {( ), ( ), ( )}, ( ). Thus, ( H ). Sce there s o choce of colors for each vertex H, H s uquely -colorable, the color classes are {0,, }, {, }, {, 5} We also show that H s a uquely -colorable graph by usg ts chromatc polyomal. Repeatedly usg () (), we have P( H, ) P( H e, ) P( H / e, ) (where e (, )) ( P( H e ) e, ) P( H e ) / e, )) ( P( H / e ) e, ) P( H / e ) / e, )) where e (0, ) e (, )) ( P(( H e e ) e, ) P(( H e e ) / e, )) ( P(( H / e ) e e, ) P(( H / e e / e, ))) P((( H / e ) / e ) e, ) P((( H / e / e ) / e, )) (where e (, )) ( ( )( ) ( ) ( ( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( ) ( )( ). Hece, P( H, ) 0 0 0 0 0!, H s uquely -colorable. The proof of Theorem (a) goes as follows. For ay teger, let P be the followg (smple) path of legth : P 0 be the complemet of P the complete graph 0,,, ( ). Thus, G G cotas two complete subgraphs wth vertces, amely, 0,,, t,, ( ) K,,, (t ),, ( ).
Uquely N-colorable Chromatcally Equvalet Graphs 9 Clearly, ( G ). We clam that G ca be colored by colors,,,, wth color dfferece. We color ( ), ( ),, (k ) ( ),, ( ) ( ). k,,, }, 0( ) p,,, t, ( k ) ( k ) Sce N 0) {(k ) ( ) for k. Smlarly, sce ( k N t) { p ) ( ) for ( p k t for, t,, }, t( ) for t,,,. Also, sce t ( p N ) {( p )( )} for p,,, k )( ) for ( k,,,, ( ) ( ). Thus, ( G ). Sce there s o choce of colors for each vertex G s uquely -colorable, the color classes are G, { 0,}, {, },, {k,k },, {, }. The proof of Theorem (b) goes as follows. For ay teger, let H be the graph wth vertces cosstg of a ( ) -cycle a tragle,.e., V ( H ) {0,,, } E ( H ) {(0, ), (, ),, (, ),,, 0), (,)}, H be the complemet of 0,,,. Thus, H ( H the complete graph cotas two complete subgraphs wth vertces, amely, 0,,, t,, ( ),,, (t ),, ( ). Clearly, ( ). We clam that H ca be colored by colors,,, H ( ), ( ),, (k ) ( k ),, ( ) ( ),,, }, 0( ) wth color dfferece. We color. Sce N 0) {(k ) ( ) for k. Smlarly, sce ( k N t) { p ) ( ) for p,,, t, k ) ( ) for k t, t ( p ( k,, }, t( ) for t,,,. Also, sce N ) { p ) ( ) t ( p for p,,,, k ) ( ) for k,,, }, ( ). Thus, ( k ( H ). Sce there s o choce of colors for each vertex H, H s uquely -colorable, the color classes are { 0,, }, {, },, {k, k ),, {, }. Followg Osterwel's dea about -clque rgs [], we let be the complete graph wth m vertces where m. For, we say that C C are adacet, deoted by C, f every vertex C s adacet to all vertces C. C C
98 C.Y. Chao For each teger ( 0, let P be the graph wth V ( P ) V ( C ) E P ) E( C C ). Thus, P has m m vertces m ( P ) m E has ( ) ( ) edges. Let 0 m m 0 P the complete graph. The we have: K m 0 G be the complemet of Corollary (a). G s a uquely -colorable graph wth m vertces. Proof. Let N be the ull graph wth m vertces for 0,,,. The proof s smlar to the proof of Theorem (a) by replacg the vertces by E N for 0,,,. For each teger G, let H be the graph wth V ( H ) V ( C ) ( H C 0 0 ) ( E( C C )) ( E( C C0 )) ( E( C )), H be the complemet of H the complete graph K q where q m. The we have: Corollary (b). H s a uquely -colorable graph wth q vertces. Proof. Let N be the ull graph wth m vertces for 0 0,,,. The proof s smlar to the proof of Theorem (b) by replacg the vertces H by N for 0,,,. Example. Let be the graph our Example, Q be the graph wth 8 V ( ) V ( ) {, } E( ) E( ) {(, ), (, )} {(, 0,,,,, 5, }. Thus, s the followg graph: ) for Q 8 G 5
Uquely N-colorable Chromatcally Equvalet Graphs 99 Sce Q cotas a complete graph wth vertces, ( ). We clam that 8 s uquely -colorable. Sce, the vertex G s a uquely -colorable graph wth colors, Q has to be colored by a ew color. Sce 8 N ( ) {( ), ( ), ( )}, ( ). Sce there s o choce of colors for each vertex Q 8, Q 8 s uquely -colorable, the color classes are: Q 8 {0, }, {, }, {, 5, }, {}. Let R 8 be the graph wth V ( ) V ( ) {, } E( ) E( ) {(, ), (, ), (, 5)} {(, ) for 0,,,,, 5}. Thus, R 8 s the followg graph: R 8 G 5 Sce R 8 cotas a complete graph wth vertces, ( R 8 ). We clam that R 8 s uquely -colorable. Sce s a uquely -colorable graph wth colors,, ( ). Sce there s o choce of colors for each vertex R 8, R 8 s uquely -colorable, the color classes are {0, }, {, }, {, 5}, {,}. We clam that R are chromatcally equvalet. By usg () (), we have 8 P(, ) P( (5, ), ) P( (5, ), ),.e., P(, ) P(, ) P( G 5, ) () G 5
00 C.Y. Chao P(, ) P( (, ), ) P( (, ), ),.e., P(, ) P( G 5, ) P( G 5, ) () Sce the polyomals o the rght sdes of () () are the same, P Q, ) P( R, ). Sce the degree of vertex s o vertex R s of ( 8 8 8 degree, are chromatcally equvalet. By usg some of the propertes of chromatc polyomals ( []), we have P(, ) ( P(, ) (( ) )) ( )( )( ) ( ) ( )( )( ) ( ) ( )( )( ) P R, ) ( P( G, ) ( ) ( 8 [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ] ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )) Thus, P(, ) P(, ). Also, P(, ) P(, )!,.e., are uquely -colorable graphs, they are chromatcally equvalet. Smlarly, we may use the graph our Example to costruct graphs Q H 9 ( 9 ( 9 H R as follows. Let V Q ) V ( H ) {, 8}, E Q ) E( ) {(,), 9 (, ), (, 5),} {(8, ) for 0,, }. {(, ), (, )} {(8, ) Also, let V R ) V ( H ) {, 8}, E R ) E( ) {(, ), (, ), (, 5)} {(8, ) ( 9 ( 9 H 0,,, }. The P ( Q9, ) ( P ( H, ) (( ) ), P( R9, ) ( P ( H, ) ) ( ). Clearly, R are ot somorphc. We ca show that Q9 9 ( Q9, ) P( R9, ) P( Q9, ) P( R9, ) P!,.e., Q9 R9 are uquely -colorable, they are chromatcally equvalet. The proof of Theorem (a) goes as follows. The graph G Theorem (a) s a uquely -colorable graph wth vertces cotag two complete subgraphs wth vertces, amely, 0,,, t,, ( ) for,,, (t ),, ( ).
Uquely N-colorable Chromatcally Equvalet Graphs 0 We costruct a graph Q wth V ( Q ) V ( G ) {, }, E Q ) E( G ) { (, ) for,,, (t ),, ( ) } {(, ) ( 0,,, }. G for Sce s uquely -colorable the vertex s cdet wth every vertex except the vertex ( ) ( ), () ( ). Sce the vertex s cdet wth every vertex for 0,,,, ( ) ( ). Thus, Q s uquely ( ) - colorable wth the color classes. { 0, }, {, },, {k,k },,{,, }, { }. Let R be the graph wth V ( R ) V ( G ) {, }, E R ) E( G ) {(, ) for,,, (t ),,( )} {(, ) for ( 0,,,( )}. Sce G s uquely -colorable the vertex s cdet wth every vertex K ()( ). Sce the vertex s cdet wth every vertex,, ( ) ( G ). Thus, R s uquely ( ) - colorable wth the color classes. { 0, }, {, },, {k, k },,{, }, {, }. We clam that Q R are ot somorphc. I Q, the degree of vertex s, R, oe of the vertces s of degree. Hece, Q R are ot somorphc. We clam that R are chromatcally equvalet. By usg (), we have Q P( Q, ) P( Q (, ), ) P( Q (, ), ), P( R, ) P( R (, ), ) P( R (, ), ). Sce P( Q (, ), ) P( R (, ), ) P( Q (, ), ) P( R (, ), ), P( Q, ) P( R, ). Hece, Q R are chromatcally equvalet, are uquely ( )- colorable. The proof of Theorem (b) goes as follows. The graph H Theorem (b) s a uquely -colorable graph wth vertces cotag two complete subgraphs wth vertces, amely, 0,,, t,, ( ),,, (t ),, ( ).
0 C.Y. Chao E Let Q be the graph wth V ( Q) V ( H ) {, } ) E( H ) {(, ) for,,, (t ),, ( )} {(, ) ( Q for 0,,, }, R be the graph wth V ( R) V ( H ) {, } E( R ) E( H ) {(, ) for,,,(t ),, ( )} {(, ) for 0,,, }. Smlar to the proof of Theorem (a), R are uquely ( ) - colorable wth the color classes. Q { 0,, }, {, },, {k, k },,{,, }, { } Q, { 0,, }, {, },, {k, k },,{, }, {, } R. Also, smlar to the proof Theorem (a), R are ot somorphc, Q P ( Q, ) P( R, ). Hece, Q are R chromatcally equvalet, are uquely ( )- colorable. I the corollary (a) (b), both of complete" subgraphs K wth G H cota two "geeralzed V ( K ) { N, N s a ull graph wth m vertces for 0,,,k,,( )}, E( K ) { N N ;,, 0,,, k,,( )}, V ( K ) { N ; N s a ull graph wth m vertces for 0,,,k,,( )}, E( K ) { N N ;,,,, (k ),, ( )}. Corollary (a). Let G N 0, N,, N be the graph corollary (a), Q be the graph wth V ( Q ) V ( G ) { N, N } where N N are ull graphs wth m vertces respectvely, E Q ) E G ) { N m ( N for,,, (t ),, ( )} 0,,, } R be the graph wth V R ) V ( G ) ) E( G ) { N ( ( { N N for { N, N } E( R N for,,, ( t ),, ( )} { N N for 0,,, }. The Q R are uquely ( )- colorable, are chromatcally equvalet.
Uquely N-colorable Chromatcally Equvalet Graphs 0 Corollary (b). Let H N 0, N,, N be the graph corollary (b), Q be the graph wth V ( Q ) V ( H ) { N, N } where N N are ull graphs wth m m vertces respectvely, E( Q ) E( H ) N N for,,, (t ),, { ( )} { N N for 0,,, }, R be the graph wth V ( R V ( H ) { N, N } E( R ) E( H ) { N N for,,, (t ),, ( )} { N N for 0,,, }. The R are uquely ( ) - colorable are chromatcally equvalet. Q Proof. (a) It s smlar to the proof of Theorem (a) by replacg the vertces R the Theorem (a) by N for 0,,,. (b) It s smlar to the proof of Theorem (b) by replacg the vertces by N for 0,,,. R Q Q Refereces. C.Y. Chao Z. Che, O uquely -colorable graphs, Dscrete Math. (99), -.. G.L. Cha, O graphs uquely colorable uder the acto of ther automorphsm groups, Dscrete Math. (99), 8-8.. R.C. Read, A troducto to chromatc polyomals, J. Comb. Theory (98), 5-.. L.J. Osterwel, Some classes of uquely -colorable graphs, Dscrete Math. 8 (9), 59-9.