Irregularity strength of regular graphs

Transcription

1 Irregularity stregth of regular graphs Jakub Przyby lo AGH Uiversity of Sciece a Techology Al. Mickiewicza 30, Kraków, Pola przybylo@wms.mat.agh.eu.pl Submitte: Nov 1, 007; Accepte: Ju 9, 00; Publishe: Ju 13, 00 Mathematics Subject Classificatios: 05C7 Abstract Let G be a simple graph with o isolate eges a at most oe isolate vertex. For a positive iteger w, a w-weightig of G is a map f : E(G) {1,,..., w}. A irregularity stregth of G, s(g), is the smallest w such that there is a w-weightig of G for which e:u e f(e) e:v e f(e) for all pairs of ifferet vertices u, v V (G). A cojecture by Fauree a Lehel says that there is a costat c such that s(g) + c for each -regular graph G,. We show that s(g) < Cosequetly, we improve the results by Frieze, Goul, Karoński a Pfeer (i some cases by a log factor) i this area, as well as the recet result by Cuckler a Lazebik. Keywors: irregularity stregth, graph weightig, regular graph 1 Itrouctio All graphs we cosier are simple a fiite. A ege {u, v} will be eote by uv or vu for short at times. For a give graph G a its vertex v, N G (v) G (v), V (G), E(G) a δ(g) (or simply N(v), (v), V, E a δ) eote the set of eighbours a the egree of v i G, the set of vertices, the set of eges a the miimum egree of G, respectively. By G[D] we mea a iuce subgraph of G with the vertex set D V (G). A set V = {V 1, V,..., V k } of isjoit subsets of a set V is calle a partitio of V if the uio of all elemets of V is V a V i for every i. We shall eote as P k a path of legth k 1 a write P k = v 1 v... v k for short if v i v i+1 are its cosecutive eges, i = 1,,..., k 1. For a graph G a a fiite set S of itegers, a S-weightig of G is a assigmet f : E(G) S. If S = {1,,..., w}, the we call f a w-weightig of G. Moreover, f(e) is calle the weight of a ege e E(G), while the weight of v V (G) is efie as f(v) = u N(v) f(vu). A weightig f is irregular if the obtaie weights of all vertices are ifferet. The smallest positive iteger w for which there exists a irregular w-weightig the electroic joural of combiatorics 15 (00), #R 1

2 of G is calle the irregularity stregth of G a is eote by s(g). If it oes ot exist, we write s(g) =. It is easy to see that s(g) < iff G cotais o isolate eges a at most oe isolate vertex. The otio of the irregularity stregth was itrouce by Chartra at al. [3]. It was motivate by the well kow fact that a simple graph of orer at least must cotai a pair of vertices with the same egree. O the other ha, a multigraph ca be irregular, i.e. the egrees of its vertices ca all be istict. Now suppose we wat to multiply the eges of a graph G i orer to create a irregular multigraph of it. The s(g) is equal to the smallest maximum multiplicity of a ege i such a multigraph, see [7] for a survey by Lehel o this parameter. We will focus our attetio o the regular graphs, which (ot oly by the ame) seem to be the most ifficult to be mae irregular. A simple coutig argumet, see e.g. [3], shows that s(g) + 1 for all -regular graphs,, of orer. A questio whether maybe just a few more weights tha this lower bou woul always suffice was pose by Jacobso (see [7]) after obtaiig a umber of supportig argumets. This was formulate as a cojecture by Fauree a Lehel. Cojecture 1 ([5]) There exists a absolute costat c such that for each -regular graph G,, of orer. They also showe the followig. s(g) + c (1) Theorem ([5]) Let G be a -regular graph,, of orer. The s(g) + 9. () About 15 years later a sizeable step forwar i the survey o this problem was mae by Frieze, Goul, Karoński a Pfeer. Theorem 3 ([6]) Let G be a -regular graph of orer with o isolate vertices or eges. (a) If (/ l ) 1/4, the s(g) 10/ + 1, (b) If (/ l ) 1/ /, the s(g) 4/ + 1, (c) If 1/ + 1, the s(g) 40(log )/ + 1. Their result was recetly supplemete (a improve i some cases) by Cuckler a Lazebik. Theorem 4 ([4]) Let G be a -regular graph of orer with o isolate vertices or eges. If 10 4/3 /3 log 1/3, the s(g) 4/ + 6. Ufortuately, these results o ot cofirm eve a weaker form of Cojecture 1, amely that s(g) c 1 + c (3) the electroic joural of combiatorics 15 (00), #R

3 hols for all -regular graphs of orer, with c 1 a c beig absolute positive costats. I other wors, we o ot eve kow if s(g) is of orer suggeste i this cojecture (see Theorem 3 (c)). We will show it quite briefly i the ext sectio, see Corollary 10. The we will improve the obtaie costats c 1, c by a careful costructio a prove the followig mai result of the paper i the last sectio. Theorem The 5 Let G be a -regular graph of orer with o isolate vertices or eges. s(g) < The right orer of s(g) Let g be a w-weightig of a graph G a let us efie m g = max { X : g(u) = g(v) for all u, v X}. X V (G) The mai iea of the proof of Theorem 3 relie o two steps. First the authors fou a w-weightig g with small m g a small w, e.g. w =, usig probabilistic tools. The they moifie g to a irregular assigmet by meas of the followig etermiistic lemma. Lemma 6 ([6]) Let G be a -regular graph without isolate vertices or isolate eges, a let g be a w-weightig of G. The, there exists a irregular ((3w 1)m g + 1)-weightig of G. Our approach, which will be explaie i etails later, is i a way similar. A equivalet of the first step escribe will be Corollary 11, which we prove at the begiig of the thir sectio. It will be resposible for groupig the set of vertices ito fairly small subsets of elemets with the same weight. Our mai tool will be the followig theorem by Aario-Berry, Dalal a Ree. Theorem 7 ([1]) Give a graph G a for all v V (G), itegers a v, a + v such that a v (v) a + v < (v), a a + v mi ( (v) + a v + 1, a v + 3 ), (4) there exists a spaig subgraph H of G such that H (v) {a v, a v + 1, a + v, a + v + 1} for all v V (G). Corollary Let > 0 be a iteger. There exists a set S of 4 cosecutive itegers such that give ay -regular graph G a umbers a v S, a + v := a v for each v V (G), there exists a spaig subgraph H of G such that H (v) {a v, a v + 1, a + v, a+ v + 1} for all v V (G). the electroic joural of combiatorics 15 (00), #R 3

4 Proof. The theorem is obvious for 3, so let 4. Assume first that is ot ivisible by 4 a take S := { 4,..., 4 3}. Clearly S = 4 a for a v S, a + v := a v , we have a v 4 3, 4 1 a + v a a + v 3 4 <, hece, by Theorem 7, it is eough to prove (4) for all v V (G). Note the that a + v = a v a v + a v + 3 a a+ v = a v + a v a v a v + + 1, thus (4) hols. Aalogously, if is ivisible by 4, we ca take S := {,..., 1}. 4 Let G = (V, E) be a graph a let A, B be two oempty, oitersectig subsets of V. For a give weightig f of eges of G, let f (A, B) := mi{ f(v) f(w) : v A, w B} eote the istace betwee A a B with respect to f. Moreover, let f (A) := 0 if f is costat o A or f (A) := mi{ f(v) f(w) : v, w A, f(v) f(w)} otherwise. Corollary 9 For each -regular graph G a a partitio {A 1,..., A } of its vertices, there exists a -weightig f of G such that f (A i, A j ) 1 for i j. Proof. Let G = (V, E) be a -regular graph with > 0 a let {A 1,..., A } be ay partitio of V. Let S = {s 1,..., s } be a appropriate set from Corollary, where s 1,..., s 4 are 4 (hece s 1 a v 4 cosecutive itegers. Let a v := s i 1 for each v A i, i = 1,..., for all v V ). By Corollary, there exists a spaig subgraph s 4 H of G such that H (v) {s i 1, s i 1 + 1, s i , si } =: Si for every v A i, i = 1...,. Note that sice s < s , the Si S j = for i j. Therefore, if we set f(e) = for all the eges of the subgraph H a f(e) = 1 for all the other eges of G, the f(v) f(w) 1 wheever v A i, w A j a i j (because G is a regular graph). A almost immeiate cosequece of the above corollary is the followig oe, which cofirms that (3) hols. Corollary 10 Let G be a -regular graph of orer with o isolate vertices or eges. The s(g) < (5) Proof. Take ay partitio {A 1,..., A } of V (G) such that A i for all i (it exists sice ( ) ). The, by Corollary 9, there is a -weightig f of G such that m f max A i, 1 i hece, by Lemma 6, we have s(g) 5 ( ) + 1 < This corollary alreay improves i may cases the results by Frieze at al., as well as the oe by Cuckler a Lazebik, see Theorems 3 a 4. the electroic joural of combiatorics 15 (00), #R 4

5 3 Improvig the upper bou i (5) The rest of the paper is evote to stregtheig the iequality (5) above, i.e. replacig costats 40 a 11 by 16 a 6. Our approach cosists also of two steps, which very roughly look as follows. First we costruct a weightig f of a give graph G that partitio the vertex set ito small subsets of vertices with the same weights, but i such a way that there is quite a big ifferece betwee the weights of vertices from istict subsets. This will be provie by Corollary 11 below, which is a immeiate cosequece of Corollary 9. The we costruct a weightig g, which is resposible for scatterig the weights of the vertices from the subsequet subsets ot too far from their iitial weights, but i such a way that as a result they all have istict weights. This is oe i Lemma 15. The sum of this two weightigs will be the esire oe. Corollary 11 For each -regular graph G a a partitio {A 1,..., A } of its vertices, there exists a weightig f : E(G) { +1, 3 +} such that f (A i, A j ) +1 for i j a f (A i ) = 0 or f (A i ) + 1 for all i. Proof. Let G = (V, E) be a -regular graph with > 0 a let {A 1,..., A } be a partitio of V. By Corollary 9, there is a -weightig h of G such that h (A i, A j ) 1 for i j. The it is eough to set f(e) = + 1 if h(e) = 1 a f(e) = 3 + if h(e) =. Note that (3 + ) ( + 1) = + 1. Therefore f(u) f(v) = 0 or f(u) f(v) +1 for u, v V, sice G is a regular graph. Cosequetly, f (A i ) = 0 or f (A i ) + 1 for each i by the efiitio of f (A i ), a f (A i, A j ) + 1 for i j by Corollary 9. Let P3 o = v 1 vv o 3 eote a path P 3 = v 1 v v 3 after removig a mile vertex v from it, but without removig ay ege. I other wors, if P 3 = (V, E) is regare as a graph (V = {v 1, v, v 3 }, E = {v 1 v, v v 3 }), the P3 o is a orere pair (V {v }, E). We shall call P3 o a ope path of legth a v o will be referre to as a ope vertex i P3 o. The other vertices of P3 o, as well as all the vertices of simple paths, e.g. P, P 3, will be calle close. We shall also abuse a little bit the establishe otatio a call P3 o a graph (or a subgraph). Now, a {P, P 3, P3 o }-factor of a graph G is a collectio of vertex (a ege) isjoit subgraphs of G which are either paths of legths 1 or, or ope paths of legth (we call them the compoets of the factor), a that together spa G. (If two graphs share oly oe vertex which is ope i oe or both of them, they are vertex isjoit.) Spa here meas that each vertex of V (G) is a close vertex of exactly oe compoet of this factor. I this sese, e.g. each star (except K 1 ) has a {P, P 3, P3 o}-factor. Let F be a forest. Deote by c F the umber of compoets of F, by L(F ) the set of leaves of F a let R(F ) = V (F ) L(F ). I orer to costruct the weightig g metioe at the begiig of this sectio (a escribe i Lemma 15) we shall ee a {P, P 3, P3 o }-factor of a give graph G cosistig of ot too may P 3 s a sufficietly may P3 o s, see Lemma 14. To obtai it, we first prove the existece of a spaig forest F of G with relatively small value of R(F ), the electroic joural of combiatorics 15 (00), #R 5

6 see Lemma 13. For this aim we shall use the omiatio umber of G, γ(g), which is the size of the smallest omiatig set of G, i.e. the subset, say D, of V (G) such that each vertex i V (G) D has a eighbour i D. The followig probabilistic result ca be fou i Alo a Specer []. Theorem 1 ([]) Let G be a graph of orer a with δ(g). The γ(g) (1 + l(δ(g) + 1)). (6) δ(g) + 1 Lemma 13 Every graph G has a spaig forest F cosistig of trees of orer at least δ(g) + 1 such that R(F ) γ(g) c F. Proof. Let D V (G) be a omiatig set of G of size γ(g) a set N v = {v} N G (v) for v D. Defie a graph H such that V (H) = {N v : v D} a N v N u E(H) iff N v N u a v u (hece N v N u sice D is the smallest omiatig set of G). Let H 1,..., H m be the coecte compoets of H a let T 1,..., T m be their respective spaig trees. Let G i = G[ N v V (H i ) N v] a D i = {v : N v V (H i )} D, i = 1,..., m. Clearly, each G i is coecte, G i δ(g) + 1, D i is a omiatig set of G i a V (G 1 )... V (G m ) = V (G), D 1... D m = D. The esire forest will cosist of spaig trees of these vertex isjoit subgraphs G i of G which we costruct i the followig maer. Take e.g. G 1. Subsequetly, for each u, v D 1 such that N u N v E(T 1 ) choose a vertex w N u N v a a to the tree the eges uw (if possible, i.e. uless u = w or uw is alreay i the tree) a vw (if possible). The we have alreay costructe a subtree of G 1 with the vertex set D 1 such that D 1 D 1 a D 1 D 1 1. Sice D 1 is a omiatig set of G 1, we ca ow joi each vertex from V (G 1 ) D 1 with a vertex from D 1 by a ege a thus costruct a spaig tree F 1 of G 1 such that R(F 1 ) D 1 1. After repeatig this process for each G i we obtai a spaig forest F (cosistig of the trees F 1,..., F m ) of G with R(F ) γ(g) c F. Lemma 14 Let G be a graph of orer a with δ(g). The there is a {P, P 3, P3 o}- factor of G cosistig of at most P δ(g)+1 3 s a with less tha 4γ(G) vertices i P s a P 3 s. Proof. Let F be a spaig forest of G with compoets F 1,..., F cf such that R(F ) γ(g) c F a F i δ(g) + 1 3, i = 1,..., c F. We process the trees F 1,..., F cf oe after aother, so let T be a arbitrary oe of them. Let u be a vertex of egree oe i this tree, where N T (u) = {w}, a let us root this tree at u. Let L 0, L 1,..., L k be the sets of vertices o the cosecutive levels of this roote tree, i.e. L i cosists of the vertices at istace i from u. The L 0 = {u}, L 1 = {w} a L k L(T ). We say that a vertex u 1 V (T ) is below (above) a vertex u V (T ) i T if u 1 (u ) lies o the path joiig u (u 1 ) with u i T a u 1 u. We will cut out the elemets of the esire factor from this tree by the followig algorithm. Process the levels of the vertices oe after aother i the reverse orer, startig at the level L k 1. O a give level, process its vertices oe after aother i a arbitrary orer. Let T 0 := T a let T i eote the tree that remais of the electroic joural of combiatorics 15 (00), #R 6

7 T i 1 after processig the cosecutive vertex. At the momet we start processig a vertex, the oly vertices left above it i the tree are its eighbours. Assume ow that we have just create T j a v V (T ) is the ext vertex to be processe. Deote by X = {x 1,..., x p } the set of eighbours of v i T j that are above v (hece X cosists exclusively of leaves of T j ). The cut off X P o 3 s of the form x l v o x l+1 from T j (by removig the vertices x l, x l+1 a the eges x l v, x l+1 v from T j ) oe after aother a iclue them as the compoets of the factor that we wat to create. If there is still a vertex i X, say x p, cut off x p v (a remove the ege joiig v with its eighbour below) as oe P to the factor. The oly exceptio to that last rule occurs if v = w (a T is o), whe istea of aig x p w, we a P 3 = x p wu to the factor. Clearly, each P a P 3 of the create {P, P 3, P3 o }-factor of T must cotai at least oe vertex from R(T ). Sice there is at most oe P 3 i this factor, these P s a P 3 may cotai at most R(T ) + 1 vertices. By repeatig this process for all F i we create a {P, P 3, P3 o }-factor of G with at most ( R(F i ) + 1) = R(F ) + c F (γ(g) c F ) + c F < 4γ(G) 1 i c F vertices i P s a P 3 s, a cosistig of at most c F i = 1,..., c F, the c F. δ(g)+1 P 3 s. Sice F i δ(g) + 1 for Lemma 15 Let G be a -regular graph of orer, 5, a let L = {,..., }. The there is such a L-weightig g of G that the obtaie vertex weights are all i L (g(v ) L) a either of the vertex weights appears more tha times (mg ). Proof. Let G be a -regular graph of orer, 5, a let L + = {1,..., }, hece L + =. Note that 4. Let us fi a {P, P 3, P3 o }-factor of G which satisfies the thesis of Lemma 14. Let A, B, C be the sets of P s, P 3 s, P3 o s, respectively, from this factor. Deote a := A, b := B a c := C, hece a + 3b + c =. By Lemma 14, b a a + 3b 4γ(G). Therefore, by (6), +1 c γ(g) + l( + 1)) (1. (7) + 1 Note that f() := 1 + l( + 1) 1 4 0, () + 1 sice f is a icreasig fuctio for > 0 a f(5) > 0 (f(5) 0, 01). By (7), () a the fact that c is a iteger, we have c. (9) Set g(e) = 0 for each ege e of G outsie the factor. Now we will weight the eges of the graphs of the factor oe after aother. Each time we weight a ege, we establish the the electroic joural of combiatorics 15 (00), #R 7

8 fial weight of at least oe (close) vertex. To esure that, for each P3 o from C, its two eges must be weighte by a pair of weights (j, j) L L, so that the weight of the ope vertex remaie uchage. First we eal with the graphs from B. If b is o (i particular, if b = 1), weight the eges of the first P 3 by 1 a 1, hece establish the weights of its three (close) vertices as 1, 0 a 1. The, oe after aother, alterately assig the pair of weights ( + i, + i) a ( i, i), i = 0, 1,,..., to the pairs of eges of the cosecutive P 3 s from B. This way the vertices of the give P 3 will obtai weights + i, + i, 3 + i or i, i, 3 i. Note that for b we have cosequetly. Moreover, sice > > b 1, (10) + 1 b, the i 1. Therefore, the establishe so far weights of vertices are all ifferet. Deote the set of these weights by U. Note also that if u U the u U. Fially, by (10), for b, 3 + ( 1) 5( + 1) = < hece i all cases we get U L. Now we weight the eges of some part of the graphs from C. Subsequetly, for the elemets of C, set weights (i, i) to the pairs of their eges (establishig the weights of their close vertices as i a i) either for all i U L + if is eve or for i L + U if is o (by (9), there is eough elemets i C). This way, each vertex weight from the set L {0} ca still be use a eve umber of times (up to the total of ). Now we weight subsequetly all P s from A. First, alterately set 1 a 1 as the weights of the eges from A (each time establishig the weights of two vertices as 1 or 1) util there is at most vertices with establishe weight 1 (a at most vertices with weight 1). The, alterately set a as the ege weights of the elemets from A util there is at most vertices weighte with. Cotiue so (with 3, 4,...) util all the eges i A have bee weighte. At this poit a weight i is establishe for the same umber of vertices as i, for i L +, with oe possible exceptio - for o a, whe for oe j L + two more vertices carry the weight j tha j. Therefore, we may easily fiish the weightig of the elemets from C. Subsequetly, for the remaiig graphs i C, set weights (i, i), i L +, to the pairs of their eges as log as possible, i.e. as log as the umber of vertices with the establishe weight i is less tha for some i L + a as log as there are still some elemets of C left uweighte. At the e either all the eges are alreay weighte a the weightig obtaie complies with our requiremets or there are still some elemets of C left uweighte. I the seco case however, by our costructio, we must have alreay, the electroic joural of combiatorics 15 (00), #R

9 weighte at least vertices (this may occur oly if a is o). Therefore, at most oe P3 o from C remaie uweighte. The we may weight its eges with 0, establishig the weight of the two remaiig vertices as 0. This way, at most 3 vertices (together with at most oe from the first part of the proof cocerig B) have weight 0. Sice 3, the costructio is complete. Proof of Theorem 5. Let G be a -regular graph,, of orer (hece < ). Assume first that 5. The by Theorem, s(g) = = = ( 5) + 7( ) + < Let ow > 5. The, by Lemma 15, there is such a weightig g of G with umbers from the set L = {,..., } that the obtaie vertex weights are all i L a either of the vertex weights appears more tha times. Let A1,..., A be a partitio of V (G) such that g(u) g(v) if u, v A i a u v, i = 1,...,. Now, by Corollary 11, there is a weightig f : E(G) { + 1, 3 + } such that f (A i, A j ) + 1 for i j a f (A i ) = 0 or f (A i ) + 1 for all i. It is easy to see that a weightig f + g is irregular for G. Therefore, sice (f + g) : E(G) {1,..., 4 + }, we have s(g) 4 + < Ackowlegemets. I wish to express my thaks to Felix Lazebik for brigig the mai problem of this paper to my attetio urig a iterestig iscussio i Buapest, a to Igo Schiermeyer for his help a remarks i Freiberg. I also eclose greetigs for the aoymous referee for their valuable commets a suggestios. The author iforms that his research were supporte by MNiSzW grat o. N N Refereces [1] L. Aario-Berry, K. Dalal, B.A. Ree, Degree costraie subgraphs, Proceeigs of GRACO005, volume 19 of Electro. Notes Discrete Math., Amsteram (005) (electroic), Elsevier. [] N. Alo, J.H. Specer, The Probabilistic Metho, Joh Wiley a Sos, Ic., 199. [3] G. Chartra, M.S. Jacobso, J. Lehel, O.R. Oellerma, S. Ruiz, F. Saba, Irregular etworks. Proc. of the 50th Aiversary Cof. o Graph Theory, Fort Waye, Iiaa (196). [4] B. Cuckler, F. Lazebik, Irregularity Stregth of Dese Graphs, to appear i J. Graph Theory. the electroic joural of combiatorics 15 (00), #R 9

10 [5] R.J. Fauree, J. Lehel, Bou o the irregularity stregth of regular graphs, Colloq Math Soc Jańos Bolyai, 5, Combiatorics, Eger North Holla, Amsteram (197) [6] A. Frieze, R.J. Goul, M. Karoński, F. Pfeer, O Graph Irregularity Stregth, J. Graph Theory 41 (00) o [7] J. Lehel, Facts a quests o egree irregular assigmets, Graph Theory, Combiatorics a Applicatios, Willey, New York (1991) the electroic joural of combiatorics 15 (00), #R 10