Technische Universität Ilmenau Institut für Mathematik

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1 Techische Uiversität Ilmeau Istitut für Mathematik Preprit No. M 07/09 Domiatio i graphs of miimum degree at least two ad large girth Löwestei, Christia; Rautebach, Dieter 2007 Impressum: Hrsg.: Leiter des Istituts für Mathematik Weimarer Straße Ilmeau Tel.: Fax: ISSN xxxx-xxxx

2 Domiatio i Graphs of Miimum Degree at least Two ad large Girth Christia Löwestei ad Dieter Rautebach 2 Dr.-Aroldi-Str. 6, D-56 Wiige, Germay, christia@wiige.de 2 Istitut für Mathematik, TU Ilmeau, Postfach 00565, D Ilmeau, Germay, dieter.rautebach@tu-ilmeau.de Abstract We prove that for graphs of order, miimum degree δ 2 ad girth g 5 the domiatio umber γ satisfies γ. As a corollary this implies that g for cubic graphs of order ad girth g 5 the domiatio umber γ satisfies γ which improves recet results due to Kostochka ad Stodolsky A upper g boud o the domiatio umber of -vertex coected cubic graphs, mauscript 2005 ad Kawarabayashi, Plummer ad Saito Domiatio i a graph with a 2-factor, J. Graph Theory , -6 for large eough girth. Furthermore, it cofirms a cojecture due to Reed about coected cubic graphs Paths, stars ad the umber three, Combi. Prob. Comput , for girth at least 8. Keywords domiatio umber; miimum degree; girth; cubic graph Itroductio The domiatio umber γg of a fiite, udirected ad simple graph G = V, E is the miimum cardiality of a set D V of vertices such that every vertex i V \ D has a eighbour i D. This parameter is oe of the most well-studied i graph theory ad the two volume moograph [4, 5] provides a impressive accout of the research related to this cocept. Fudametal results about the domiatio umber γg are upper bouds i terms of the order ad the miimum degree δ of the graph G. Ore [0] proved that γg 2 provided δ. For δ 2 ad all but 7 exceptioal graphs Blak [] ad McCuaig ad Shepherd [9] proved γg 2. Equality i these two bouds is attaied for ifiitely 5 may graphs which were characterized i [9,, 6]. I [] Reed proved that γg for every graph G of order ad miimum degree 8 at least ad he cojectured that this boud could be improved to for coected cubic graphs. While Reed s cojecture was disproved by Kostochka ad Stodolsky [7] who costructed a sequece G k k N of coected cubic graphs with lim k γg k V G k + 69,

3 Kostochka ad Stodolsky [8] proved γg 4 for every coected cubic graph G of order > 8 ad γg + 8 g 2 for every coected cubic graph G of order > 8 ad girth g where the girth is the legth of a shortest cycle i G. The last result improved a recet result due to Kawarabayashi, Plummer ad Saito [6] who proved that γg + 2 9k + for every 2-edge coected cubic graph G of order ad girth at least k for some k N. The first to use the girth g of a graph G ext to its order ad miimum degree δ to boud the domiatio umber γ were probably Brigham ad Dutto [2] who proved γ 2 g 6 provided that δ 2 ad g 5. I [4, 5] Volkma determied fiite set of graphs G i for i {, 2} such that γ 2 g 6 i + 6 uless G is a cycle or G G i. Motivated by these results Rautebach [2] proved that for every k N there is a fiite set G k of graphs such that if G is a graph of order, miimum degree δ 2, girth g 5 ad domiatio umber γ that is ot a cycle ad does ot belog to G k, the γ 2 g 6 k. I the preset paper we prove a best-possible upper boud o the domiatio umber of graphs of miimum degree at least 2 ad girth at least 5 which allows to improve ad 2 for large eough girth. Furthermore, it cofirms Reed s cojecture [] for cubic graphs with girth at least 8. 2 Results We immediately proceed to our mai result. Theorem If G = V, E is a graph of order, miimum degree δ 2, girth g 5 ad domiatio umber γ, the γ g+. + 2

4 Proof: For cotradictio, we assume that G = V, E is a couterexample of miimum sum of order ad size. Let, g ad γ be as i the statemet of the theorem. Sice ad γ 2 are liear with respect to the compoets of G ad is o-decreasig i g, the g+ + graph G is coected. Furthermore, the set of vertices of degree at least is idepedet. We prove several claims restrictig the structure of G. Claim. G has a vertex of degree at least. Proof of Claim : For cotradictio, we assume that G has o vertex of degree at least. I this case G is a cycle of order at least g ad γ =. If = g, the, if g 0 mod, = < +2 = + < g+ g g+2, if g mod ad, if g 2 mod. If = g +, the = +2 = + < < g+ g g+2, if g 0 mod,, if g mod ad, if g 2 mod. Fially, if g, the. g Sice g + g +, if g 0 mod, + = g, if g mod ad g, if g 2 mod, we obtai i all cases the cotradictio γ ad the proof of the claim g+ + is complete. A path P i G betwee vertices x ad y of degree at least whose iteral vertices are all of degree 2 will be called 2-path ad we set p P x := y ad p P y := x. Claim 2. G has o two vertices u ad v of degree at least that are joied by a 2-path P of legth mod. Proof of Claim 2: For cotradictio, we assume that such vertices u ad v ad such a path P exist.

5 If V deotes the set of iteral vertices of the path, the G[V ] is a path of order 0 mod which has a domiatig set D of cardiality. Sice the graph G[V \ V ] satisfies the assumptios of the theorem, we obtai, by the choice of G, that G[V \ V ] has a domiatig set D of cardiality at most V g+ +. Now, D D is a domiatig set of G ad we obtai γ D + D + g+ + < g+, + which implies a cotradictio ad the proof of the claim is complete. Claim. G has o vertex u of degree at least that lies o a cycle C of legth mod whose vertices differet from u are all of degree 2. Proof of Claim : For cotradictio, we assume that such a vertex u ad such a cycle C exist. Let V deote a miimal set of vertices cotaiig a eighbour of u o the cycle C such that G[V \ V ] has o vertex of degree less tha 2. If u is of degree at least 4, the the graph G[V ] is a path of order 0 mod ad we obtai the same cotradictio as i Claim 2. Hece we ca assume that u is of degree. I this case the graph G[V ] arises from C by attachig a path to u. Sice G[V ] has a spaig subgraph which is a path, it has a domiatig set D of cardiality at most As before, G[V \V ] has a domiatig set D with D. g+ Now D D is a domiatig set of G ad usig g we obtai γ D + D + = g+ g+ + + V + g Cosiderig the three cases = g, V = g + ad = g as i the proof of Claim implies the cotradictio γ ad the proof of the claim is g+ + complete. 4.

6 Claim 4. G has o vertex u of degree at least that lies o two cycles C ad C 2 of legths 2 mod whose vertices differet from u are all of degree 2. Proof of Claim 4: For cotradictio, we assume that such a vertex u ad such cycles C ad C 2 exist. Let V deote a miimal set of vertices cotaiig a eighbour of u o the cycle C ad a eighbour of u o the cycle C 2 such that G[V \ V ] has o vertex of degree less tha 2. If u is of degree at least 6, the the graph G[V ] cosists of two disjoit paths of order mod whose edvertices are adjacet to u. This easily implies that there is a set D {u} V cotaiig u such that every vertex i V \ D has a eighbour i D ad D =. Sice g, we obtai a similar cotradictio as i the proof of Claim. Hece we ca assume that u is of degree at most 5. I this case the graph G[V ] cosists of C ad C 2 ad possibly a path attached to u. Agai, it is easy to see that G[V ] has a domiatig set D of cardiality at most. Sice g, we obtai a similar cotradictio as i the proof of Claim ad the proof of the claim is complete. Claim 5. G has o two distict vertices u ad v of degree at least such that u lies o a cycle C of legth 2 mod whose vertices differet from u are all of degree 2, ad u ad v are joied by a 2-path P of legth 2 mod. Proof of Claim 5: For cotradictio, we assume that such vertices u ad v, such a cycle C ad such a path P exist. Let V deote a miimal set of vertices cotaiig a eighbour of u o the cycle C ad a eighbour of u o the path P such that G[V \ V ] has o vertex of degree less tha 2. If u is of degree at least 5, the the graph G[V ] is the uio of two paths of order mod which both have a edvertex that is adjacet to u. Agai, there is a set D {u} V cotaiig u such that every vertex i V \ D has a eighbour i D ad D =. Sice g, we obtai a similar cotradictio as i the proof of Claim. Hece we ca assume that u is of degree at most 4. Let P deote the 2-path startig at u that is iterally disjoit from C ad P. Let w deote the edvertex of P differet from u, i.e. w = p P u. If v w or v = w ad v is of degree at least 4, the the graph G[V ] arises from C, P ad P by deletig v ad w. If v = w ad v is of degree, the let P deote the 2-path startig at v that is iterally disjoit from P ad P. Now the graph G[V ] arises from C, P, P ad P by deletig the edvertex of P differet from v. I both cases, by the parity coditios, the graph G[V ] has a domiatig set D of. Sice g, we obtai a similar cotradictio as i the proof cardiality at most of Claim ad the proof of the claim is complete. Claim 6. G has o vertex u that is joied to three vertices v, v 2 ad v of degree at least by three distict 2-paths of legths 2 mod. Proof of Claim 6: For cotradictio, we assume that such vertices u, v, v 2 ad v ad such paths exist. Let P, P 2 ad P deote the three 2-paths joiig u to v, v 2 ad v, 5

7 respectively. Let V 0 deote the set of iteral vertices of the three paths ad let V deote a miimal set of vertices cotaiig V 0 such that G[V \V ] has o vertex of degree less tha 2. I order to complete the proof of Claim 6, we isert aother claim about the structure of G[V ]. Claim 7. If u, v, v 2, v, P, P 2, P, V 0 ad V are as above, the i either u V ad G[V ] is the uio of three paths of order mod each of which has a edvertex that is adjacet to u, ii or G[V ] has a spaig subgraph which arises by idetifyig a edvertex i each of three or four paths of which three are of order 2 mod, iii or g ad G[V ] has a spaig subgraph which arises by idetifyig a edvertex i each of three or four paths of which two are of order 2 mod, iv or u V, g ad G[V ] has a spaig subgraph which is the uio of three paths each of which has a edvertex that is adjacet to u ad two of these three paths are of order mod. Proof of Claim 7: If w is a vertex of degree at most i G[V \ V 0], the let P w deote the 2-path startig i w that is iterally disjoit from V 0. Note that P w has legth 0 if w is a isolated vertex i G[V \ V 0]. First, we assume that {v, v 2, v } =, i.e. the vertices v, v 2 ad v are all distict. If u is of degree, the V = {u} V 0 ad ii holds. If u is of degree at least 5, the V = V 0 ad i holds. Hece we ca assume that u is of degree 4. If either p P u u {v, v 2, v } or p P u u {v, v 2, v }, say pu = v, ad v is ot of degree, the ii holds. Hece we ca assume that pu = v is of degree. Let P deote the 2-path startig i v that is iterally disjoit from V 0 ad P u. If either p P v {v 2, v } or p P v {v 2, v }, say p P v = v 2, ad v 2 is ot of degree, the ii holds. Hece we ca assume that p P v = v 2 is of degree. Let P deote the 2-path startig i v 2 that is iterally disjoit from V 0 ad P. If either p P v 2 v or p P v 2 = v ad v is ot of degree, the ii holds. Hece we ca assume that p P v 2 = v is of degree. Let P deote the 2-path startig i v that is iterally disjoit from V 0 ad P. Clearly, p P v {u, v, v 2 } ad ii holds. Note that we ca delete the edges icidet to v i i P i for i i order to obtai the spaig subgraph metioed i ii. Next, we assume that {v, v 2, v } =. Note that the 2-paths betwee u ad v = v 2 = v form cycles of legth at least g. 6

8 If u ad v are both of degree at least 5, the V = V 0 ad i holds. If u is of degree at most 4 ad v is of degree at least 5, the ii holds. Note that if v V, the we ca delete the edges icidet to v i P i for i i order to obtai the spaig subgraph metioed i ii. If u is of degree at least 5 ad v is of degree at most 4, the ii holds. Note that if u V, the we ca delete the edges icidet to u i P i for i i order to obtai the spaig subgraph metioed i ii. If u ad v are both of degree at most 4, the either P u = P v ad ii holds or P u P v ad iii holds. Note that i the last case we ca delete the edges icidet to v i P ad P 2 i order to obtai the spaig subgraph metioed i iii. Fially, we assume that {v, v 2, v } = 2, say v = v v 2. Note that the 2-paths P ad P betwee u ad v = v form a cycle of legth at least g. If v is of degree at least 4, the we ca argue similarly as i the case {v, v 2, v } =. Hece we ca assume that v is of degree. If u ad v are joied by a 2-path Q differet from P ad P, the iii or iv hold depedig o the degree of u. Note that, if u is of degree four for istace, the we ca delete the edge icidet to u i Q ad the edge icidet to v i P i order to obtai the spaig subgraph metioed i iii. Hece we ca assume that u ad v are ot joied by a 2-path differet from P ad P. If u is of degree 4 ad u ad v 2 are joied by a 2-path differet from P 2, the iii holds. Hece we ca assume that either u is of degree at least 5 or u ad v 2 are ot joied by a 2-path differet from P 2. I the remaiig cases iii or iv hold which completes the proof of the claim. We retur to the proof of Claim 6. Note that i Cases i or iv of the Claim 7 there is a set D {u} V cotaiig u such that every vertex i V \ D has a eighbour i D ad either D Case i or D ad g Case iv. Furthermore, by the parity coditios, i Cases ii ad iii of Claim 7, the graph G[V ] has a domiatig set D such that either D Case ii or D ad g Case iii. As before, G[V \V ] has a domiatig set D with D V g+ + ad D D is a domiatig set of G. If D, the we obtai a similar cotradictio as i Claim 2 ad if D ad g, the we obtai a similar cotradictio as i Claim. This completes the proof of the claim. We have by ow aalysed the structure of G far eough i order to describe a sufficietly small domiatig set leadig to the fial cotradictio. Let V deote the set of vertices of degree at least ad let = V. The graph G[V \ V ] is a collectio of paths of order either mod or 2 mod. 7

9 Let P, P 2,..., P s deote the set of vertices of the paths of order mod ad let Q, Q 2,..., Q t deote the set of vertices of the paths of order 2 mod. By the above claims, which implies t 2 s + t 2 ad ad s s 2t. For i s, the path G[P i ] without its oe or two edvertices has a domiatig set Di P of cardiality P i. For j t, the path G[Q j ] without its two edvertices has a domiatig set D Q j of cardiality Q j 2. Now the set s t V i= D P i is a domiatig set of G ad we obtai, s γ + D P i + i= j= t j= D Q j D Q j s P i t = + + i= j= = s 2t s +. + s i= P i + t This fial cotradictio completes the proof. j= i= Q j Q j 2 P i + t Note that Theorem is best possible for the uio of cycles C. We derive some g+ + cosequeces of Theorem for graphs of miimum degree at least. Corollary 2 If G = V, E is a graph of order, miimum degree δ, girth g 5 ad domiatio umber γ, the γ g+ 4α G 4 + α G 4 + where α G 4 deotes the idepedece umber of G 4, i.e. the maximum cardiality of a set I V of vertices such that every two vertices i I are at distace at least 5. 8 j= Q j

10 Proof: Let I V be a set of vertices such that every two vertices i I are at distace at least 5 ad I = α G 4. If V = I N G I, the 4 I. We will prove that G[V \V ] has miimum degree at least 2. Therefore, for cotradictio, we assume that there is a vertex u V \V which has 2 eighbours v ad v 2 i V. Clearly, v N G w ad v 2 N G w 2 for some w, w 2 I. If w = w 2, the uv w v 2 u is a cycle of legth 4 which is a cotradictio. If w w 2, the w v uv 2 w 2 is a path of legth 4 betwee two vertices of I which is a cotradictio to the choice of I. Therefore, G[V \ V ] has miimum degree at least 2 ad, by Theorem, it has a domiatig set D with D set of G ad we obtai which completes the proof. g+ +. Now I D is a domiatig γg I + D 4 V + g+ + α G 4 + g+ 4α G 4 + Sice αg for every graph G of order ad maximum degree ad the maximum + degree of G 4 is at most 2 2 2, we obtai the followig immediate corollaries. Corollary If G = V, E is a cubic graph of order, girth g 5 ad domiatio umber γ, the 44 γ g Proof: If g 2, the ad Reed s boud [] implies the desired result. If 5 5g 8 g, the G 4 is either complete or a odd cycle ad Brooks theorem [] implies that α G 4 G 4 45 ad the result follows from Corollary 2. Note that < for g 8 ad hece Corollary improves the bouds ad 5 5g 2 due to Kostochka ad Stodolsky [8] ad Kawarabayashi, Plummer ad Saito [6] ad also cofirms Reed s cojecture [] for large eough girth. Corollary 4 For every δ there are costats α δ, < ad β δ, such that if G = V, E is a graph of order, miimum degree δ, maximum degree, girth g 5 ad domiatio umber γ, the γ α δ, + β δ,. g Istead of givig exact expressios for α δ, ad β δ, i Corollary 4, we pose it as a ope problem to determie the best-possible values for these coefficiets. 9

11 Refereces [] M. Blak, A estimate of the exteral stability umber of a graph without suspeded vertices, Prikl Math i Programmirovaie Vyp 0 97, -. [2] R.C. Brigham ad R.D. Dutto, Bouds o the domiatio umber of a graph, Q. J. Math., Oxf. II. Ser , [] R.L. Brooks, O colourig the odes of a etwork, Proc. Cambridge Philos. Soc. 7 94, [4] T.W. Hayes, S.T. Hedetiemi ad P.J. Slater, Fudametals of domiatio i graphs, Marcel Dekker, Ic., New York, 998. [5] T.W. Hayes, S.T. Hedetiemi ad P.J. Slater, Domiatio i graphs advaced topics, Marcel Dekker, Ic., New York, 998. [6] K. Kawarabayashi, M.D. Plummer ad A. Saito. Domiatio i a graph with a 2-factor, J. Graph Theory , -6. [7] A.V. Kostochka ad B.Y. Stodolsky, O domiatio i coected cubic graphs, Discrete Math , [8] A.V. Kostochka ad B.Y. Stodolsky, A upper boud o the domiatio umber of -vertex coected cubic graphs, mauscript [9] W. McCuaig ad B. Shepherd, Domiatio i graphs with miimum degree two, J. Graph Theory 989, [0] O. Ore, Theory of graphs, Amer. Math. Soc. Colloq. Publ. 8, 962. [] B. Raderath ad L. Volkma, Characterizatio of graphs with equal domiatio ad coverig umber, Discrete Math , [2] D. Rautebach, A Note o domiatio, girth ad miimum degree, to appear i Discrete Math. [] B. Reed, Paths, stars ad the umber three, Combi. Prob. Comput , [4] L. Volkma, Upper bouds o the domiatio umber of a graph i terms of diameter ad girth, J. Combi. Math. Combi. Comput , -4. [5] L. Volkma, A upper boud for the domiatio umber of a graph i terms of order ad girth, mauscript [6] B. Xu, E.J. Cockaye, T.H. Hayes, S.T. Hedetiemi ad S. Zhou, Extremal graphs for iequalities ivolvig domiatio parameters, Discrete Math , -0. 0