γ-max Labelings of Graphs

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1 γ-max Labeligs of Graphs Supapor Saduakdee 1 & Varaoot Khemmai 1 Departmet of Mathematics, Sriakhariwirot Uiversity, Bagkok, Thailad Joural of Mathematics Research; Vol. 9, No. 1; February 017 ISSN E-ISSN Published by Caadia Ceter of Sciece ad Educatio Correspodece: Supapor Saduakdee, Departmet of Mathematics, Sriakhariwirot Uiversity, Sukhumvit 3, Bagkok, Thailad. aa o rr@hotmail.com Received: November 11, 016 Accepted: December 5, 016 Olie Published: Jauary 3, 017 doi: /jmr.v91p90 Abstract URL: Let G be a graph of order ad size m. A γ-labelig of G is a oe-to-oe fuctio f : V(G) {0, 1,,..., m} that iduces a edge-labelig f : E(G) {1,,..., m} o G defied by The value of f is defied as The maximum value of a γ-labelig of G is defied as while the miimum value of a γ-labelig of G is f (e) = f (u) f (v), for each edge e = uv i E(G). val( f ) = e E(G) f (e). val max (G) = max{val( f ) : f is a -labelig of G}; val mi (G) = mi{val( f ) : f is a γ-labelig of G}. I this paper, we give a alterative short proof by mathematical iductio to achieve the formulae for val max (K r,s ) ad val max (K ). Keywords: γ-labelig, value of a γ-labelig 1. Itroductio Let G be a graph of order ad size m. A γ-labelig of G is defied i (Chartrad, Erwi, VaderJagt & Zhag, 005) as a oe-to-oe fuctio f : V(G) {0, 1,..., m} that iduces a edge-labelig f : E(G) {1,..., m} o G defied by f (e) = f (u) f (v) for each edge e = uv of G. The value of f is defied by val( f ) = f (e). e E(G) If the edge-labelig f of a γ-labelig f of a graph is also oe-to-oe, the f is a graceful labelig. Amog all labeligs of graphs, graceful labeligs are probably the best kow ad most studied. Graceful labeligs origiated with a paper of Rosa (Rosa, 1966), who used the term β-valuatios. A few years later, Golomb (Golomb, 197) called these labeligs graceful ad this is the termiology that has bee used sice the. Gallia (Gallia, 009) has writte a extesive survey o labeligs of graphs. The subject of γ-labeligs of graphs was studied i (Bulligto, Eroh, & Witers, 010; Chartrad, Erwi, VaderJagt, & Zhag, 005; Crosse, Okamoto, Saepholphat, & Zhag, 007; Foseca, Saepholphat, & Zhag, 013; Foseca, Khemmai, & Zhag, 015; Foseca, Saepholphat, & Zhag, 011; Khemmai & Saduakdee, 015, 016). Obviously, sice γ-labelig f of a graph G of order ad size m is oe-to-oe, it follows that f (e) 1, for ay edge e, ad therefore, val( f ) m. Moreover, G has a γ-labelig if ad oly if m 1 ad every coected graph has a γ-labelig. The maximum value ad the miimum value of a γ-labelig of G are defied i (Chartrad, Erwi, VaderJagt, & Zhag, 005) as val max (G) = max{val( f ): f is a γ-labelig of G} ad val mi (G) = mi{val( f ): f is a γ-labelig of G}, 90

2 Joural of Mathematics Research Vol. 9, No. 1; 017 respectively. A γ-labelig g of G is a γ-max labelig if val(g) = val max (G) ad a γ-labelig h is a γ-mi labelig if val(h) = val mi (G). Figure 1 shows ie γ-labeligs f 1, f,..., f 9 of the path P 5 of order 5 (where the vertex labels are show above each vertex ad the iduced edge labels are show below each edge). The value of each γ-labelig is show i Figure 1 as well. Sice val( f 1 ) = 4 ad the size of P 5 = 4, it follows that f 1 is a γ-mi labelig of P 5. It is show i (Chartrad, Erwi, VaderJagt, & Zhag, 005) that the γ-labelig f 9 is a γ-max labelig of P f 1 : f : f 3 : val( f 1 ) = 4 val( f ) = 5 val( f 3 ) = 6 f 4 : f 5 : f 6 : val( f 4 ) = 7 val( f 5 ) = 8 val( f 6 ) = 9 f 7 : f 8 : f 9 : val( f 7 ) = 10 val( f 8 ) = 10 val( f 9 ) = 11 Figure 1. Some γ-labeligs of P 5 By the γ-spectrum of a graph G, we mea the set spec(g) = {val( f ) : f is a γ-labelig of G}. Observe that val mi (G), val max (G) spec(g) for every graph G. For itegers a ad b with a < b, let be a cosecutive set of itegers betwee a ad b. Thus for every graph G, The spa of a γ-labelig f of a graph G is defied as [a, b] = {a, a + 1,..., b} spec(g) [val mi (G), val max (G)]. spa( f ) = max{ f (v) : v V(G)} mi{ f (v) : v V(G)}. Cosequetly, if G P 5, the spec(g) = {4, 5, 6, 7, 8, 9, 10, 11} ad for each γ-labelig f of G, spa( f ) = 4 0 = 4. For a γ-labelig f of a graph G of size m, the complemetary labelig f : V(G) {0, 1,..., m} of f is defied by f (v) = m f (v) for v V(G). Not oly is f a γ-labelig of G as well but val( f ) = val( f ). This gives us the followig. Observatio 1 (Chartrad, Erwi, VaderJagt, & Zhag, 005) Let f be a γ-labelig of a graph G. The f is a γ-max labelig (γ-mi labelig) of G if ad oly if f is a γ-max labelig (γ-mi labelig) of G. The followig result appeared i (Foseca, Khemmai, & Zhag, 015) is useful to us. Theorem 1 (Foseca, Khemmai, & Zhag, 015) If f is a γ-max labelig of a otrivial graph G of order ad size m, the {0, m} f (V(G)). I (Bulligto, Eroh & Witers, 010; Chartrad, Erwi, VaderJagt, & Zhag, 005), the maximum ad miimum values of a γ-labelig of path P, cycle C, complete graph K, double star S p,q ad complete bipartite graph K r,s are determied. For ay positive itegers, m, with m, let G be a otrivial graph of order ad size m, a γ -labelig of G is defied i (Foseca, Saepholphat, & Zhag, 013) as a oe-to-oe fuctio f : V(G) {0, 1,..., m, m + 1,..., } that 91

3 Joural of Mathematics Research Vol. 9, No. 1; 017 iduces a edge-labelig f : E(G) {1,,..., } o G defied by f (e) = f (u) f (v) for each edge e = uv of G. The value of f is defied by val( f ) = f (e). The maximum value of a γ -labelig of G is e E(G) val max(g) = max{val( f ): f is a γ -labelig of G}. The miimum value of a γ -labelig of G is val mi (G) = mi{val( f ): f is a γ -labelig of G}. A γ -labelig g of G is a γ -max labelig if val(g) = val max(g) ad a γ -labelig h is a γ -mi labelig if val(h) = val mi (G). Note that val max (G) = val max(g) ad val mi (G) = val mi (G) whe = m. We first make the followig observatio for γ -max ad γ -mi labeligs of graphs. Observatio Let f be a γ -labelig of a graph G. The f is a γ -max labelig (γ -mi labelig) of G if ad oly if f is a γ -max labelig (γ -mi labelig) of G. I 010, the explicit formula for val max (K r,s ) ad the stadard form for γ-max labelig of complete bipartite graph K r,s were determied by Bulligto, Eroh ad Witers (Bulligto, Eroh, & Witers, 010). Later, Foseca, Khemmai ad Zhag (Foseca, Khemmai, & Zhag, 015) i 015 preseted the alterative proof of the formula for val max (K r,s ) that employs γ-mi labeligs of complete graphs, as we state ext. Theorem (Bulligto, Eroh & Witers, 010) For ay two positive itegers r s, val max (K r,s ) = rs (rs 1 ) (r + s) + 1. Theorem 3 (Bulligto, Eroh, & Witers, 010) Let f be a γ-labelig of complete bipartite graph K r,s with partite sets V r ad V s of cardiality r ad s, respectively. The f is a γ-max labelig of K r,s if ad oly if 1. f (V r ) = [0, r 1] ad f (V s ) = [rs (s 1), rs], or. f (V r ) = [rs (r 1), rs] ad f (V s ) = [0, s 1]. I 005, the maximum value of γ-labelig of complete graph K was determied by Chartrad et al. (Chartrad, Erwi, VaderJagt, & Zhag, 005). As well, the authors (Foseca, Khemmai, & Zhag, 015) characterized the γ-max labelig of complete graph K i 015. Theorem 4 (Chartrad, Erwi, VaderJagt, & Zhag, 005) For every positive iteger, ( 1)(3 5+6) 4 if is odd val max (K ) = ( ) 4 if is eve. Theorem 5 (Foseca, Khemmai, & Zhag, 015) Let f be a γ-labelig of a complete graph K. The f is a γ-max labelig of K if ad oly if [ ] [( 0, 1 ) ( + 1, )] if is eve f (V(K )) = [ ] [( 0, 1 ) ( + 1, )] {k} if is odd, where k [, ( ) ]. The goal of this paper is to preset a alterative approach to formulae for val max (K r,s ) ad val max (K ) proved by mathematical iductio. The reader is referred to Chartrad ad Zhag (Chartrad & Zhag, 005) for basic defiitios ad termiology ot metioed here. 9

4 Joural of Mathematics Research Vol. 9, No. 1; 017. γ -max Labeligs of Graphs I this sectio, we begi our ivestigatio for γ -max labelig of ay otrivial graph by presetig a useful lemma. Lemma 1 Let f be a γ -max labelig of a otrivial graph G of order ad size m. Let u, w V(G) with f (u) = mi{ f (v): v V(G)} ad f (w) = max{ f (v): v V(G)}. The eighborhoods of u ad w are ot empty. Proof. For ay otrivial coected graph G, it is obvious that N(u) ad N(w) are ot empty. Let G be a discoected graph. We will show that N(u) 0 ad N(w) 0 Assume, to the cotrary, that N(u) = 0 or N(w) = 0. Case 1. N(u) = 0. The u is a isolated vertex of G. Sice G has a γ -max labelig, m 1. Therefore, there is a compoet G 1 of G with V(G 1 ). Let x V(G 1 ) with f (x) = mi{ f (v): v V(G 1 )}. Let g be a γ -labelig of G defied by The which is a cotradictio. val(g) = val( f ) f (x) if v = u g(v) = f (u) if v = x f (v) if v u, x. v N(x) = val( f ) v N(x) ( f (v) f (x)) + v N(x) ( f (v) f (x)) + = val( f ) + N(x) ( f (x) f (u)) > val( f ), v N(x) (g(v) g(x)) ( f (v) f (u)) Case. N(w) = 0. By a similar argumet, this leads to a cotradictio with the maximum value of a γ -labelig of G. We ow show formula for spa of γ -max labeligs of graphs. Propositio 1 Let G be a otrivial graph of order ad size m ad f a γ -labelig of G. If f is a γ -max labelig of G, the spa( f ) =. Proof. Let f be a γ -max labelig of G. Let u, w V(G) with f (u) = mi{ f (v): v V(G)} ad f (w) = max{ f (v): v V(G)}. The f (u) 0 ad f (w). Assume, to the cotrary, that spa( f ) <. The f (w) f (u) <. Therefore, f (u) > 0 or f(w) > 0. Case 1. f (u) > 0. Let g be a γ -labelig of G defied by The which is a cotradictio. g(v) = val(g) = val( f ) v N(u) = val( f ) v N(u) 0 if v = u f (v) if v u. ( f (v) f (u)) + v N(u) ( f (v) f (u)) + = val( f ) + N(u) ( f (u) 0) > val( f ) (by Lemma 1), v N(u) (g(v) g(u)) ( f (v) 0) Case. f(w) > 0. A similar argumet to the oe used i Case 1 leads to a cotradictio with the maximum value of a γ labelig of G. This also provides the followig corollary. Corollary 1 Let G be a otrivial graph of order ad size m ad f a γ -labelig of G. If f is a γ -max labelig of G, the {0, } f (V(G)). 93

5 Joural of Mathematics Research Vol. 9, No. 1; γ-max Labeligs of Complete Bipartite Graphs We defie the γ -spectrum of a graph G by spec (G) = {val( f ): f is a γ -labelig of G}. Cosequetly, {val mi (G), val max(g)} spec (G) for every graph G. As a illustratio, we ow establish the γ -spectrum of a star K 1,s. Propositio For positive itegers s, with s, {( ) k + 1 spec (K 1,s ) = ( s + 1 ) + ( ) } k + 1 : 0 k. Proof. Let K 1,s be a star with V(K 1,s ) = {v} V s where v is a cetral vertex ad V s = {v 1, v,..., v s } ad f a γ -labelig of a graph K 1,s with f (v) = k where 0 k. If k = 0, the we may assume that f (v i ) = (s i) for all 1 i s. The val( f ) = If k =, the by Observatio, s f (v i ) f (v) = i=1 If 0 < k <, the we may assume that Therefore, as desired. f (v i ) = s ( ) ( ) + 1 s + 1 ( (s i)) =. i=1 ( ) ( ) + 1 s + 1 val( f ) =. i 1 if 1 i k (s i) if k + 1 i s. val( f ) = (k + (k 1) + + 1) + (( (s 1)) + ( (s )) + + ( k)) = ( ) ( k+1 + k+1 ) ( s+1 ), I Propositio, we cosidered γ -spectrum of a star K 1,s. We are ow ready to compute the maximum value of a γ -labelig of K 1,s. Corollary For positive itegers s, with s, ( ) ( ) + 1 s + 1 val max(k 1,s ) =. Moreover, let f be a γ -labelig of K 1,s with The f ad f are oly γ -max labeligs of K 1,s. f (v) = 0 ad f (V s ) = [ (s 1), ]. Next, we show a alterative ad yet simple proof employig mathematical iductio of Theorem 3 which is proposed by Bulligto, Eroh ad Witers (Bulligto, Eroh & Witers, 010) i 010 ad by Foseca, Khemmai ad Zhag (Foseca, Khemmai & Zhag, 015) i 015. I order to do this, first, let K r,s be a complete bipartite graph with partite sets V r ad V s of cardialities r ad s, respectively, where V r = {u 1, u,..., u r } ad V s = {v 1, v,..., v s } ad the we discuss γ -max labeligs of K r,s as follows. Theorem 6 Let f be a γ -labelig of a complete bipartite graph K r,s with f (V r ) = [0, r 1] ad f (V s ) = [ (s 1), ], where rs. The f ad f are oly two γ -max labeligs of K r,s. 94

6 Joural of Mathematics Research Vol. 9, No. 1; 017 Proof. We proceed by iductio o r + s. The result is certaily true for r + s =. Assume that r + s 3 ad the result holds for K r,s whe r + s < r + s. By Corollary, hece the theorem holds whe r = 1. Suppose that r. Let f be a γ -max labelig of K r,s with f (u 1 ) < f (u ) < < f (u r ) ad f (v 1 ) < f (v ) < < f (v s ). Assume that f (u 1 ) < f (v 1 ). By Corollary 1, f (u 1 ) = 0. Furthermore, for each j {1,,..., s} it follows that f (v j ) (s j). Let K r 1,s be a complete bipartite graph with vertex set V(K r 1,s ) = V(K r,s ) {u 1 } ad partite sets V r 1 = V r {u 1 } ad V s. Cosequetly, let f 1 be a γ 1 -labelig of K r 1,s defied by The f 1 (u) = f (u) 1 for each u V(K r 1,s ). val( f 1 ) = i r 1 j s f 1 (u iv j ) val 1 max(k r 1,s ). Let g 1 be a γ 1 -max labelig of K r 1,s. Sice 1 (r 1)s, by iductio hypothesis, we have g 1 (V r 1 ) = [0, r ] ad g 1 (V s ) = [( 1) (s 1), ( 1)]. We ca exted g 1 to a γ -labelig g of K r,s defied by 0 if u = u 1 g(u) = g 1 (u) + 1 otherwise. Sice it follows that val max(k r,s ) = val( f ) = f (u i v j ) + s f (v j ) f (u 1 ) i r 1 j s val( f 1 ) + s (s j) 0 val 1 max(k r 1,s ) + s (s j) 0 = val(g 1 ) + s (s j) 0 = val(g) val max(k r,s ), ad From (1), we have s f (v j ) f (u 1 ) = From (), we have f 1 (V r 1 ) = [0, r ], hece ad we have f (u 1 ) = 0. Therefore, s (s j) 0 (1) val( f 1 ) = val 1 max(k r 1,s ). () f (V s ) = [ (s 1), ]. f (V r 1 ) = [1, r 1] f (V r ) = [0, r 1] ad f (V s ) = [ (s 1), ]. O the other had, if f (v 1 ) < f (u 1 ), the a similar argumet to the oe used shows that f (V s ) = [0, s 1] ad f (V r ) = [ (r 1), ]. 95

7 Joural of Mathematics Research Vol. 9, No. 1; 017 The followig result is the cosequece of Theorem 6 whe = rs. Theorem 7 Let K r,s be a complete bipartite graph with partite sets V r ad V s of cardialities r ad s, respectively, let f be a γ-labelig of K r,s with f (V r ) = [0, r 1] ad f (V s ) = [rs (s 1), rs]. The f ad f are oly two γ-max labeligs of K r,s. 4. γ-max Labeligs of Complete Graphs The γ-max labeligs of complete graphs K were characterized i (Foseca, Khemmai & Zhag, 015). I this sectio, we preset characterizatio of γ -max labeligs ad γ-max labelig of complete graphs K, by applyig a similar fashio to the oe used i the proof of Theorem 6. Theorem 8 Let f be a γ -labelig of a complete graph K with [ ] [ ] 0, 1 + 1, f (V(K )) = [ ] [ ] 0, 1 + 1, {k} if is eve if is odd where ( [ ] ) ad k,. The f ad f are oly two γ -max labeligs of K. Proof. Let K be a complete graph with V(K ) = {u 1, u,..., u }. Assume that is eve. We use mathematical iductio o. Whe =, the result is obvious. Assume that 4 ad the result holds for K whe is eve ad <. Let f be a γ -max labelig of K with f (u 1 ) < f (u ) < < f (u ). By Corollary 1, f (u 1 ) = 0 ad f (u ) =. Let f 1 be a γ -labelig of a complete graph K with vertex set V(K ) = {u, u 3,..., u 1 } defied by f 1 (u i ) = f (u i ) 1 for each i 1. Let g 1 be a γ -max labelig of K. Sice ( ), by iductio hypothesis, we have ] g 1 (V(K )) = [ 0, We ca exted g 1 to a γ -labelig g of K defied by 1 [ ( ) 0 if u = u 1 g(u) = if u = u g 1 (u) + 1 if u u 1, u. ] + 1, ( ). Sice it follows that Thus, Hece val max(k ) = val( f ) = val( f 1 ) + 1 ( f (u i ) f (u 1 )) + 1 ( f (u ) f (u i )) + ( f (u ) f (u 1 )) val i= i= max(k ) + 1 ( 0) + ( 0) i= = val(g 1 ) + 1 (g(u i ) g(u 1 )) + 1 (g(u ) g(u i )) + (g(u ) g(u 1 )) = val(g) i= val max(k ), f 1 (V(K )) = f (V(K )) = [ 0, [ 1, i= val( f 1 ) = val max(k ). ] [ 1 ( ) ] [ ( ) ] + 1, ( ). ] +, ( 1) 96

8 Joural of Mathematics Research Vol. 9, No. 1; 017 ad we have f (u 1 ) = 0, f (u ) =. Therefore, f (V(K )) = [ ] [ ] 0, 1 + 1,. O the other had, if is odd, the by a similar argumet, this shows that [ ] [ ] f (V(K )) = 0, 1 + 1, {k} where k [, ]. The followig result is the cosequece of Theorem 8 whe = ( ). Theorem 9 Let f be a γ-labelig of a complete graph K with [ ] [( 0, 1 ) ( + 1, )] if is eve f (V(K )) = [ ] [( 0, 1 ) ( + 1, )] {k} if is odd where k [ (, ] ). The f ad f are oly two γ-max labeligs of K. 5. Ope Questio The characterizatio of γ-max labeligs of K r,s ad K were determied. The mai ope questio is to characterize γ-mi labeligs of those graphs. Refereces Bulligto, G. D., Eroh, L. L., & Witers, S. J. (010). γ-labeligs of complete bipartite graphs. Discuss. Math. Graph Theory, 30, Chartrad, G., Erwi, D., VaderJagt, D. W., & Zhag, P. (005). γ-labeligs of graphs. Bull. Ist. Combi. Appl, 44, Chartrad, G., Erwi, D., VaderJagt, D. W., & Zhag, P. (005). O γ-labeligs of trees. Discuss. Math. Graph Theory, 5(3), Chartrad, G., & Zhag, P. (005). Itroductio to Graph Theory. The Walter Rudi studet series i advaced mathematics, McGraw-Hill Higher Educatio, Bosto. Crosse, L., Okamoto, F., Saepholphat, V., & Zhag, P. (007). O γ-labeligs of orieted graphs. Math. Bohem, 13, Foseca, C. M., Saepholphat, V., & Zhag, P. (013). Extremal values for a γ-labelig of a cycle with a triagle. Utilitas Math, 9, Foseca, C. M., Khemmai, V., & Zhag, P. (015). O γ-labeligs of graphs. Utilitas Math, 98, Foseca, C. M., Saepholphat, V., & Zhag, P. (011). The γ-spectrum of a graph. Ars Combi, 101, Gallia, J. A. (009). A dyamic survey of graph labelig. Electro. J. Combi, 16(6), Golomb, S. W. (197). How to umber a graph, I: Graph Theory ad Computig, Academic Press, New York. Hegde, S. M. (000). O (k, d)-graceful graphs. J. Combi. Iform. System Sci, 5, Khemmai, V., & Saduakdee, S. (015). The γ-spectrum of cycle with oe chord. Iteratioal Joural of Pure ad Applied Mathematics, 105(4), Rosa, A. (1966). O certai valuatios of the vertices of a graph, I: Theory of Graphs (Iterat. Sympos., Rome, , Gordo ad Breach, New York, Duod, Paris. Saduakdee, S., & Khemmai, V. (016). γ-labelig of a cycle with oe chord. Discrete ad Computatioal Geometry ad Graphs, Lecture Notes i Computer Sciece, 9943, Copyrights Copyright for this article is retaied by the author(s), with first publicatio rights grated to the joural. This is a ope-access article distributed uder the terms ad coditios of the Creative Commos Attributio licese ( 97