On (K t e)-saturated Graphs

Transcription

1 Noame mauscript No. (will be iserted by the editor O (K t e-saturated Graphs Jessica Fuller Roald J. Gould the date of receipt ad acceptace should be iserted later Abstract Give a graph H, we say a graph G is H-saturated if G does ot cotai H as a subgraph ad the additio of ay edge e E(G results i H as a subgraph. I this paper, we costruct (K 4 e-saturated graphs with E(G either the size of a complete bipartite graph, a 3-partite graph, or i the iterval [ 4, ] + 6. We the exted the (K4 e-saturated graphs to (K t e-saturated graphs. Keywords K 4 e K t e saturated edge spectrum 1 Itroductio The study of saturated graphs has see a recet surge i popularity. A graph G is H-saturated if, give a graph H, G does ot cotai a copy of H but the additio of ay edge e E(G creates at least oe copy of H withi G. The questio of the miimum umber of edges of a H-saturated graph o vertices, kow as the saturatioumber ad deoted sat(, H, has bee addressed for may differet types of graphs. The saturatioumber cotrasts the popular questio of the maximum umber of edges possible i a graph G o vertices that does ot cotai a copy of H, kow as the Turáumber ad deoted ex(, H. Now, the topic of iterest is the problem of fidig the edge spectrum for H-saturated graphs. The edge spectrum of the family of H- saturated graphs o vertices is the set of all possible sizes of a H-saturated graph. For terms ot defied here see [4]. Jessica Fuller Departmet of Mathematics ad Computer Sciece, Emory Uiversity, Atlata, GA. jmfulle@mathcs.emory.edu Roald J. Gould Departmet of Mathematics ad Computer Sciece, Emory Uiversity, Atlata, GA. rg@mathcs.emory.edu

2 Jessica Fuller, Roald J. Gould We are iterested, the, i costructig graphs of size m that are H- saturated with sat(, H m ex(, H ad determiig if such a graph exists for every possible value of m. This has bee explored for few graphs H, icludig K 3, K t ad P t. The spectrum for K 3 -saturated graphs was foud i 1995 by Barefoot, Casey, Fisher, Fraghaugh, ad Harary [3]. I [1], Ami, Faudree, ad Gould foud the spectrum for K 4 -saturated graphs ad i [] Ami, Faudree, Gould ad Sidorowicz foud the spectrum for K t, t 4. Cotiuig this work, Gould, Tag, Wei, ad Zhag addressed the edge spectrum of small paths [6]. Oe goal of this paper is to determie the edge spectrum of (K 4 e- saturated graphs, where K 4 e is the complete graph o four vertices with oe edge removed. The graph K 4 e is isomorphic to the graph comprised of two triagles that share a edge, sometimes called a book. Further, sat(, K 4 e = 3( 1 (see [5]. The saturatioumber, sat(, K 4 e, ca be realized as the edge cout of the graph o vertices formed by 1 triagles joied at a sigle vertex v whe is odd (Figure 1(a ad triagles joied at the vertex v with a edge from v to the remaiig vertex whe is eve (Figure 1(b. These graphs are (K 4 e-saturated as each vertex, except perhaps oe, is a vertex of a triagle ad a additioal edge creates a secod triagle with v, formig a copy of (K 4 e. v v (a Figure 1 (b The Turáumber for a vertex (K 4 e-free graph G is ex(, K 4 e = ad ca be realized by the complete bipartite graph K,. The goal ow is to costruct graphs of size m that are (K 4 e-saturated with 3( 1 m ad to determie if such a graph exists for every possible value of m. Proof of Lower Boud We begi with some useful lemmas. Lemma 1 If G is a coected (K 4 e-saturated graph, the diam(g =. Proof Suppose that G is a coected (K 4 e-saturated graph. Let x, y V (G with xy E(G. The G+xy cotais a K 4 e so there is a vertex w V (G, distict from x ad y such that x, w, y is a path i G. Sice this must be true for ay pair of vertices i G, diam(g =.

3 O (K t e-saturated Graphs 3 Lemma If G is a (K 4 e-saturated graph o vertices with a cut vertex, the E(G =. 3( 1 Proof Let G be a (K 4 e-saturated graph with a cut vertex, say x. By Lemma 1, diam(g = so every such path from u to v is of legth, that is x is adjacet to every vertex y V (G x. Sice it is possible to add a edge betwee two vertices of degree oe without creatig a copy of (K 4 e ad G is (K 4 e-saturated, there is a maximal matchig i V (G x that covers all 3( 1 except possibly oe vertex. This creates edge disjoit triagles, with oe additioal edge icidet to x if is eve. This is precisely the graph that realizes the saturatioumber with a edge cout of E(G =. 3( 1 Aside from the saturatioumber, small edge couts are ot realizable by (K 4 e-saturated graphs. The followig lemmas show the lower boud o the edge spectrum of (K 4 e-saturated graphs. Lemma 3 Let G be a coected (K 4 e-saturated graph with miimum degree δ(g 3 o 10 vertices. The E(G 4. Proof Let G be a coected (K 4 e-saturated graph with miimum degree δ(g 3. If δ(g 4, the E(G > 4. Therefore there exists a vertex of degree exactly 3, say u. Note that diam(g = by Lemma 1. Let u be adjacet to exactly three other vertices of G, say x, y ad z. Let X = {x, y, z} ad let A = V (G {u, x, y, z}. Sice diam(g =, every vertex i A is adjacet to at least oe of the vertices i X. Let A 1 be the set of vertices i A that are adjacet to exactly oe vertex of X, let A be the vertices i A adjacet to exactly two vertices of X ad let A 3 be the vertices i A adjacet to all vertices of X. The miimum degree coditio implies that each v A 1 must be adjacet to at least two other vertices i A ad each w A must be adjacet to at least oe other vertex i A. So we have a miimum edge cout as follows: A A1 + A E(G 3 + A 1 + A + 3 A A 1 + A + A + 3 A 3 = 3 + ( A A 3 = 5 + A + A 3. If either A or A 3 is o-empty, we are doe. Thus, assume that A = A 3 = 0. The E(G 5 ad it remais to show that there is at least oe additioal edge i G. If at least oe of the edges xy, yz, xz is i E(G, we are doe. Assume that xy, yz, ad xz are ot edges of G. Sice δ(g = 3, there must be at least two vertices of A 1 adjacet to x, two vertices of A 1 adjacet to y ad two vertices

4 4 Jessica Fuller, Roald J. Gould of A 1 adjacet to z. The each vertex adjacet to x must be adjacet to at least oe vertex adjacet to y ad at least oe vertex adjacet to z. Each vertex adjacet to y must be adjacet to at least oe vertex adjacet to x ad oe vertex adjacet to z. Each vertex adjacet to z must be adjacet to at least oe vertex adjacet to x ad oe vertex adjacet to y. This requiremet allows the miimum possible edge cout to remai at E(G 5 as it requires at least A 1 edges amogst the vertices of A 1. However, this graph is ot (K 4 e-saturated, as addig xy does ot create a copy of K 4 e, so there must be at least oe additioal edge. This completes the proof of the lemma. Lemma 4 Let G be a -coected (K 4 e-saturated graph o m edges ad 10 vertices. The m 4. Proof Let G be a (K 4 e-saturated, -coected graph o m edges. Sice G is (K 4 e-saturated, diam(g = by Lemma 1 ad it follows from Lemma 3, that m 4 if δ(g 3. Suppose δ(g = with deg(z = for some z V (G. The z is adjacet to some x, y V (G ad we ca partitio the remaiig vertices of G ito three sets A, B, C with A N(x, B N(x N(y ad C N(y, (see Figure. Sice G is -coected, A ad B caot both be empty, as this would make y a cut vertex. Similarly, C ad B caot both be empty. Note that if B the edge from x to y is ot i E(G as it would create a copy of K 4 e ad for similar reasos, B must be a idepedet set. z x y A B C Figure Case 1: Suppose both A ad C are empty ad B is ot empty. Each vertex i B is adjacet to both x ad y, which creates a copy of C 4 for each vertex of B with the vertices x, y ad z. Addig the edge xy, ay edge betwee vertices of B, or the edge vz for some v B will create a K 4 e, so the graph G is (K 4 e-saturated. I this case, m = + ( 3 = 4. Case : Suppose that A is empty ad B, C are o-empty. Sice diam(g =, there must be a path of legth two from x to each w C hece, there must be a edge from at least oe v B to each w C. Sice G caot cotai a copy of K 4 e, each w C must be adjacet to a distict

5 O (K t e-saturated Graphs 5 vertex i B ad hece B. The each w C is i a distict triagle ad is ot adjacet to aother vertex i C or a copy of K 4 e would exist i G. Additioal edges will icrease the edge cout so E(G must be at least m + B + = + ( 3 + = = 4. Note that by symmetry a similar argumet holds whe C is empty ad A is oempty. Case 3: Suppose that A ad C are both o-empty with 1 A ad B is empty. Sice G is (K 4 e-saturated, xy must be a edge of G ad there ca be o path of legth or more betwee ay two vertices i A or betwee ay two vertices i C. Also, diam(g = implies that there is a u w path of legth 1 or for each u A ad each w C. So each u A must be adjacet to at least vertices of C. There must also be at least additioal edges, either withi C i the form of a matchig, or betwee A ad C if there is a vertex of A that is ot i a edge i A. If = 1, diam(g = requiresthat either w C is adjacet to all vertices i A or w C is adjacet to vertices i A ad there are edges A A withi A. I either case, E(G 4+ A + A = ( 3+4 = > 4. Otherwise, we have the followig edge cout for G: m 3 + A + + A + = + = 4 + = 4 + = 4 + ( ( ( ( ( ( ( ( ( Sice 1 ad A = 3 1 clearly hold, it follows that E(G 4 is always true. Case 4: Suppose that A, B ad C are o-empty with 1 A. The diam(g = implies that there must be a path of legth at most from each u A to each w C. The vertices x ad y caot be adjacet as B is

6 6 Jessica Fuller, Roald J. Gould o-empty, which results i at least oe C 4 with v B, x, y ad z. Also, the vertices i B must be idepedet as ay edge betwee two vertices i B will result i a K 4 e with x ad y. If some u A is ot adjacet to ay vertex i A or B, the the edge uz does ot create a copy of K 4 e, hece the graph is ot (K 4 e-saturated. So if a vertex u A is idepedet withi A, the uv E(G for some v B ad either uw or vw is a edge of G for every w C. If a vertex u A is ot idepedet withi A, the uw E(G for some w C as a edge from A to B gives a K 4 e. By symmetry, the same is true for vertices i C. If = 1, diam(g = requires that there is a path of legth 1 or betwee w C ad each vertex i A. Sice G is (K 4 e-saturated, wv is a edge i G for some v B ad either vu or wu is also a edge of G for some u A. The w C is adjacet to at least A 1 vertices i A ad there are at most edges withi A or from A to B. I ay case, A 1 E(G 3 + A + B A = ( = 4. Otherwise, each u A must be adjacet to at least vertices of C. The there must also be at least additioal edges, either withi C i the form of a matchig, or betwee A ad C if there is a vertex of A that is ot i a edge i A. This yields the followig edge cout of G: m + A + B + + A + = + B + ( B 3 = + = 5 + = 5 + = 5 + ( ( B ( ( ( ( B + 3 ( B 4. ( A. ( ( ( + 4 So E(G 4 if 3 ad A. However, if A = 1 the E(G 4, similar to the case whe = 1, so it remais to determie the edge cout of G whe A = =. Suppose that C = {w, w }. If ww E(G, wv ad w v are ot edges of G for ay v B. The diam(g = implies that each u A is adjacet to w or w. Also, sice G is (K 4 e-saturated, either there is a edge i A or there is a edge from A to B as addig oe of those edges does ot create a copy of K 4 e. This yields E(G 6+ B = 10+( 7 = 4. O the other had, if ww E(G, both wv ad w v are edges of G for distict v B

7 O (K t e-saturated Graphs 7 ad v B without creatig a copy of K 4 e ad each vertex of C must be adjacet to at least oe vertex i A such that E(G 6+ B ++ = 4. This completes the proof of the lemma. 3 Proof of Theorem We will ow show that there is a (K 4 e-saturated graph for every iteger value of m i the iterval [ 4, + 6] by combiig two differet costructios. Theorem 1 There exists a (K 4 e-saturated graph o 6 vertices ad m edges where 4 m + 6. Proof Case 1: Suppose 4 m 3 9. x y x z y (a B A A C B (b Figure 3 To costruct (K 4 e-saturated graphs we modify K, ad K 3, 3. For the graph i Figure 3(a, we form a set B by removig at most 3 vertices from A so that the vertices are adjacet to x ad to distict vertices i A ad we form a set C with a sigle vertex from A so that it is adjacet to y ad a vertex i A that is ot adjacet to ay vertex i B. We the joi each vertex of B to the vertex i C. We ow show the resultig graphs are (K 4 e-saturated. First, the edge xy will create at least two triagles o that edge if A. Ay edge added withi A will create a K 4 e with x ad y ad ay edge added withi B (or withi C will create a triagle with x (or y, respectively, which creates a copy of K 4 e as every pair of vertices i B (or C is a part of two triagles. Each edge from A to B is a edge of a triagle with x so ay additioal edge betwee u A ad v B will create aother triagle with x ad the edge xv, resultig i a copy of (K 4 e. Similarly, a edge betwee ay vertex i A ad w C with create a K 4 e. Fially, ay additioal edge from x to w C or y to v B will create a (K 4 e with the triagle costructed betwee x or y ad B or C, sharig the edge from A to w or v, respectively.

8 8 Jessica Fuller, Roald J. Gould For the graph i Figure 3(a, if A = b 3 where B = b ad = 1, the edge cout is: m = A + 3 B + = ( b 3 + 3b + = 4 + b. So we have a edge cout of m = 4 + b, which icreases by oe as the size of B icreases by oe. Sice 0 B 3, we have 4 m 4 + = 5 6. For the graph i Figure 3(b, we form a set B with vertices from A so that the vertices are adjacet to x ad to distict vertices i A. Similar to the graphs i Figure 3(a, such graphs are (K 4 e-saturated ad, if A = b 3 where B = b, they have edge cout: m = 3 A + B = 3( b 3 + b = 3 9 b. So the edge cout decreases by oe as the size of B icreases by oe. Sice 0 B 3, we have 3 9 m = 5 7, which clearly itersects the iterval costructed with the graphs of Figure 3(a. Thus, we have costructed saturated graphs of size 4 to 3 9 for 6 ad this case completes the proof of Theorem 1 for 11. Case : Suppose 3 9 m 4 8. We ca similarly modify the complete bipartite graphs K 3, 3 ad K 4, 4 by addig triagles to the vertices i the smaller vertex set i the same way as before, to obtai a (K 4 e-saturated graph for 11. x y z x y z w A B A C B (a D (b Figure 4 C For the graph i Figure 4(a, we form a set B with vertices from A so that the vertices are adjacet to x ad to distict vertices i A. We form a set C with the vertices from A that are adjacet to z ad distict vertices i A ad we a form set D with vertices from A that are adjacet to y ad

9 O (K t e-saturated Graphs 9 distict vertices i A. I formig the sets B, C ad D, it is ecessary that their eighbors i A do ot overlap. We the joi each vertex of B ad C to all of D. If A = b c d 3 where B = b, = c ad D = d, the the edge cout is: m = 3 A + B + + D + B D + D = 3( b c d 3 + b + c + ( + b + cd = 3 b c + (b + c d 9. If we fix B = = 1, we have a edge cout of m = d, which icreases by oe as the size of D icreases by oe. Sice the costructio requires B + + D 3, we have D 3 so that 3 9 m = 7 3. For the graph i Figure 4(b, we form a set B with vertices from A so that the vertices are adjacet to x ad to distict vertices i A, we form a set C with the vertices from A that are adjacet to w ad distict vertices i A. The we joi each vertex i C to all vertices i B. If A = b c 4 where B = b ad = c, the the edge cout is: m = 4 A + B + + B = 4( b c 4 + b + c + bc = 4 b c + bc 6. Thus, if we fix = c = 1, we have a edge cout of m = 4 18 b, which decreases by oe as the size of B icreases by oe. The 0 B 4 implies 4 8 m 4 8 ( 4 = 7 5, which itersects the iterval for the graphs of Figure 4(a. Thus, we have costructed graphs of size 3 9 to 4 8 for 11. Case 3: Suppose 4 8 m + 5. We blow-up the graph C 5 such that each vertex becomes a set of idepedet vertices with adjacecies accordig to the origial C 5, where a edge xy E(C 5 becomes a K s,t, whe x V (C 5 blows-up to a set of s vertices ad y V (C 5 blows-up to a set of t vertices. The ay edge added withi a set of idepedet vertices will create at least two triagles o that edge with vertices of two adjacet sets. Also, ay edge added betwee vertices i two differet vertex sets will create at least two triagles o that edge with vertices of the commo adjacet set, if the commo adjacet set has order at least. As such, a blow-up C 5 i Figure 5(a, with at least two vertices i each vertex set, is (K 4 e-saturated.

10 10 Jessica Fuller, Roald J. Gould A A D C D C (a E B (b Figure 5 E B The blow-up C 5 i Figure 5(b, which we deote as G = C 5 [A, D, E, B, C], is (K 4 e-saturated with A = b c 5 provided B = b, = c, D = ad E = 3 with E(G = m give by the products of the orders of cosecutive vertex sets, hece: m = A D + D E + E B + B + A = ( b c 5 + (3 + 3b + bc + c( b c 5 = c + c 7c + b 4 = ( c(c + 5c + b 4. The for fixed values of c, whe b icreases by 1, that is, as vertices are moved from A to B, the edge cout icreases by 1. To maitai at least two vertices i each set of the blow-up C 5, we must have b [, c 7]. If we let b = c 7 for fixed, the we have m = c + 3 c 8c 1, which is maximized whe c = 4 such that c [, 4]. The the smallest edge cout for G is whe a = 9, b =, c = ad is m = ( ( = ad the largest possible edge cout is give whe a =, b = 3, c = 4 ad is: m = ( + 4 ( ( 4 + ( 3 4 = ( ( = = + 5 = + 5. Next, we check that the etire iterval [ 4 8, ] + 5 of legth 5+4 is covered. For each fixed c, we will have a iterval of values S c determied by the rage of values for b, amely, each iterval has a left edpoit give whe b = such that the iterval starts at m = ( c(c + 5c. So we have a (K 4 e-saturated graph o vertices ad m

11 O (K t e-saturated Graphs 11 edges ( for a iterval of legth ( c = c 8 ad we have = 5 such itervals. The the ext cosecutive iterval will start at: m = ( (c + 1(c (c + 1 = ( c(c + + ( c (c + 5c 7 = ( c(c + 5c + ( c 8. Thus, the ed of each iterval S c will overlap with the ext iterval S c+1 i the first ( c 8 ( c 8 = c umbers. There are 5 itervals, each with ( c 8 (c + 1 = c 7 distict elemets. As the lowest iterval starts at 4 8 ad the largest iterval eds at + 5 with each iterval havig a oempty overlap with the ext highest iterval, all values are covered. Case 4: Suppose m = + 6. Similar to the first two cases, we modify the complete bipartite K,. Let A be the first partite set ad B the secod partite set. We remove all edges from oe vertex v A ad all edges from oe vertex v B. We add a edge from v to a vertex a A ad a vertex b B, creatig a triagle. Similarly, we add a edge from v to a vertex a A ad a vertex b B, where a a ad b b. Fially, we add the edge vv. Sice this is the same modificatio as i Case 1, the resultig graph is similarly (K 4 e-saturated with a edge cout of m = ( ( + 5 = + 6. This completes the proof of the theorem. 4 Graphs i [ + 7, ] We cojecture that graphs with sizes i the iterval [ + 7, ] are of two types: complete bipartite graphs with partite sets of early equal size, ad 3-partite graphs with two partite sets of early equal size ad oe partite set of order oe. The complete bipartite graph is (K 4 e-saturated as addig a edge betwee ay two oadjacet vertices will create a K 4 e. I the 3-partite graph, we let the two larger partite sets iduce a complete bipartite graph ad the sigle vertex set be adjacet to exactly oe vertex i each of the other partite sets. This 3-partite graph, the, cotais a complete bipartite graph o vertices as well as a sigle triagle ad is (K 4 e- saturated. If a edge is added withi either of the idepedet sets a copy of K 4 e is created ad if a edge is added betwee a vertex of the triagle ad ay other vertex of the graph, a K 4 e is created. Such graphs would have the highest possible edge couts whe the larger partite sets are almost the same order. Let the graph G be a complete bipartite graph with oe partite set of order + k. The the size of G is m = ( + k ( k. This gives

12 1 Jessica Fuller, Roald J. Gould a few additioal values of m i the iterval [ + 7, ]. Let the graph H be a 3-partite graph described above with partite sets of order + k, k, ad 1. The the size of H is: m = ( + k ( k + = k + k k k +. Which is k k + for eve ad k k + for odd. For small values of k, this gives additioal values of m [ + 7, ] +. We believe that this completes the edge spectrum of (K 4 e-saturated graphs sice the graphs G ad H are completely saturated with four vertex complete graphs missig two edges. Thus, the edge spectrum of (K 4 e-saturated graphs has a jump from the saturatioumber to the ext possible edge cout, is cotiuous i the iterval [ 4, ] + 6 ad the,we believe, has sporadic values i the iterval [ + 7, ]. 5 Costructig (K t e-saturated Graphs Give a graph G that is (K 4 e-saturated, it is possible to costruct a graph G = G + v that is (K 5 e-saturated where G + v is costructed by addig a vertex v ad all edges from v to each vertex i G. The, by joiig a vertex to each of the (K 4 e-saturated graphs we have costructed, there is a (K 5 e- saturated graph o vertices for each edge cout i [ ( 4 + (, 1 = [ 3 7, 1 1 ] ( ( ] There are also (K 5 e-saturated graphs for sporadic values of m i the iterval [ , 1 1 ] +. Similarly, give a graph H that is (K t 1 e-saturated, it is possible to costruct a graph H = H +v that is (K t e-saturated where H +v is costructed by addig a vertex v ad all edges from v to each vertex i H. So joiig t 4 vertices, oe at a time, to a (K 4 e-saturated graph o t + 4 vertices will result i a (K t e-saturated graph. As such, there is a (K t e-saturated graph o vertices ad m edges for each value i the iterval [ t t 4 i=1 ( t i, t+4 t+4 ] + t + + t 4 i=1 ( t i [ = (t t + 3 t, ( t = [ (t ( t 1, t ( + t ] + ( t + (t 4 t + 7 t 4 t ( + (t 3 t ].

13 O (K t e-saturated Graphs 13 Also, there are (K t e-saturated graphs for sporadic values of m i [ t t ( + (t 3 t + 4, t t ( + (t t 1 ]. 6 Ackowledgmets We would like to thak the referees for their thorough readig ad providig exceedigly helpful commets. Refereces 1. Ami, K., Faudree, J., Gould, R. J., The edge spectrum of K 4 -saturated graphs.. Combi. Math Combi. Comput. 81(01, Ami, K., Faudree, J., Gould, R. J., Sidorowicz, E., O the o (p -partite K p free graphs. Discuss. Math. Graph Theory 33(013, o. 1, Barefoot, C., Casey, K., Fisher, D., Fraughaugh, K., Harary, F., Size i maximal triagle-free graphs ad miimal graphs of diameter. Discrete Mathematics, 138(1995, Chartrad, G. ad Lesiak, L., Graphs & Digraphs, 5th ed., Chapma & Hall/CRC, Boca Rato, FL, Che, G., Faudree, R. J., Gould, R. J., Saturatioumbers of books. Electroic Joural Combiatorics. 15(008, o. 1, Research Paper 118, 1 pp. 6. Gould, R. J., Tag, W., Wei, E., Zhag, C. Q., Edge spectrum of saturatioumbers for small paths. Discrete Mathematics. 31(01,