Average Coectivity ad Average Edge-coectivity i Graphs Jaehoo Kim, Suil O July 1, 01 Abstract Coectivity ad edge-coectivity of a graph measure the difficulty of breakig the graph apart, but they are very much affected by local aspects like vertex degree. Average coectivity (ad aalogously, average edge-coectivity has bee itroduced to give a more refied measure of the global amout of coectivity. I this paper, we prove a relatioship betwee the average coectivity ad the matchig umber i all graphs. We also give the best lower boud for the average edge-coectivity over -vertex coected cubic graphs, ad we characterize the graphs where equality holds. I additio, we show that this family has the fewest perfect matchigs amog cubic graphs that have perfect matchigs. 1 Itroductio A graph G is k-coected if it has more tha k vertices ad every subgraph obtaied by deletig fewer tha k vertices is coected. The coectivity of G, writte κ(g, is the maximum k such that G is k-coected. The coectivity of a graph measures how may vertices must be deleted to discoect the graph. However, sice this value is based o a worst-case situatio, it does ot reflect how well coected the graph is i a global sese. For example, a graph G obtaied by addig oe edge joiig two large complete graphs has the same coectivity as a tree. However, it is much easier to disturb the tree, which is relevat if they both model commuicatio systems. Departmet of Mathematics, Uiversity of Illiois, Urbaa, IL, 61801, kim805@math.uiuc.edu. research partially supported by the Arold O. Beckma Research Award of the Uiversity of Illiois at Urbaa- Champaig. Departmet of Mathematics, The College of William ad Mary, Williamsburg, VA, 3185, so@wm.edu. 1
I 00, Beieke, Oellerma ad Pippert [1] itroduced a parameter to give a more refied measure of the global amout of coectivity. The average coectivity of a graph G with vertices, witte κ(g, is defied to be κ(u,v u,v V (G, where κ(u, v is the miimum ( umber of vertices whose deletio makes v ureachable from u. By Meger s Theorem, κ(u, v is equal to the maximum umber of iterally disjoit paths joiig u ad v. Note that κ(g κ(g = mi u,v V (G κ(u, v. I Sectio, we prove a boud o the average coectivity i terms of matchig umber ad characterize whe equality holds. I Sectio 3, the average edge-coectivity of a graph G will be itroduced. We obtai a sharp lower boud o the average edge-coectivity of a coected cubic graph with vertices. We will show that equality holds oly graphs i a family. The family is also useful to fid the miimum umber of perfect matchigs over -vertex cubic graphs havig a perfect matchig. I fact, Lovász ad Plummer [7] cojectured i the 1970s that if G is a cubic graph without cut-edges, the the umber of perfect matchigs i G should grow expoetially may with the umber of vertices of G. Recetly, the cojecture was proved by Esperet, Kardos, Kig, Král, ad Nori [5]. I Sectio, we show that if weake the coditio -edge-coectedess to has a perfect matchig, the a cubic graph has to have at least perfect matchigs ad there are ifiitely may cubic graphs with perfect matchigs. Average Coectivity ad Matchig Number Regardig average coectivity, several properties are kow. The followig is oe of them. Theorem.1. (Dakelma, Oellerma, 003 [3] If G has average degree d ad vertices, ( d d κ(g d 1 We prove a boud o the average coectivity i terms of matchig umber. The matchig umber of a graph G is deoted by α (G. We first itroduce the defiitios of M-alteratig path ad M-augmetig path. Defiitio.. Give a matchig M, a M-alteratig path is a path that alterates betwee edges i M ad edges ot i M. A M-alteratig path whose edpoits are missed by M is a M-augmetig path. Theorem.3. For a coected graph G, κ(g α (G, (1
ad this is sharp oly for complete graphs with a odd umber of vertices. I additio, if G is a -vertex coected bipartite graph, the ( 9 κ(g 8 3 α (G, ( 8( 8 ad this is sharp oly for the complete bipartite graph K q,3q, where q is a positive iteger. Proof. First, we show that iequality (1 holds for ay coected graph G. Let M be a maximum matchig i G, ad let m = M. Let S = V (G V (M, s = S, ad = V (G. Note that = m + s. If s 1, the m 1, ad the boud holds sice k(g 1 m. Thus, we may assume that s. For vv M, put v ad v ito T, T, or R as follows: If either v or v has a eighbor i S, the put both i T. If v has a eighbor i S ad v does ot, the put v T ad v T. If both have eighbors i S, put them both i R. I this last case, ote that if v ad v have distict eighbors i S, the M is ot maximal. Hece each has exactly oe eighbor i S, which forms a triagle with them. We cosider three cases to obtai upper bouds o κ(u, v depedig o the possible locatios of distict vertices u ad v. Case 1: u S. First, ote that S is idepedet. Furthermore, if P ad P are distict iterally disjoit u, v-paths, the both of them must visit V (M T immediately after u. Sice P ad P have o vertex i commo, we have κ(u, v m t, where t = T. Case : u, v T. Clearly, κ(u, v 1 = m + s 1 Case 3: u R T. Recall that every vertex i R has exactly oe eighbor i S. For the vertex after u o a u, v-path, at most oe vertex of S is available. Thus, if u R T ad v V (G, the there are at most m cadidates to begi such a path sice u V (M there are m 1 other vertices i V (M, ad oe eighbor of u is i S. Let t = T. Note that by the defiitos of T ad T, we have t t. The, we have (m t (( ( ( s + s( s + (m + s 1 t ( ( + m s ( t s( s κ(g ( (m t( ( ( ( s + (m + s 1 t ( + (m tts + m( s t ts ( ( t (s 1( = m + t s t s m t s + ts + t 1 m. (3 ( 1 ( The last iequality of (3 holds because whe t 1, we have s + 3ts + t 1 0 ad whe t = 0, we have t s +3ts+t 1 ( 1 = 0. 3
To have equality i the last iequality of (3, we eed to have t = 0 or t = 1 sice if t, the t s +3ts+t 1 > 0. ( 1 Whe t = 1, equality requires s = 0, which implies that M is a perfect matchig. Thus m =. I this case, κ(g 1 < = m, which implies that we caot have equality i (3. If t = 0 ad s, the we have a bigger matchig tha M, sice every vertex i R has exactly oe eighbor i S ad G is coected. Thus, whe t = 0, equality requires s 1. If s = 0, the m =, which is bigger tha 1 κ(g. I this case, we caot have equality i (3. Thus, we have s = 1, which implies m = 1. The is odd, ad if κ(g = 1, the G = K. Thus, equality holds oly whe G is a complete graph with a odd umber of vertices. To prove that iequality ( holds, we cosider a -vertex coected bipartite graph G with partite sets A ad B. Let M be a maximum matchig i G. Let m = M, let A 1 = A V (M, ad let B 1 = B V (M. Let B be all vertices i B that are reachable by a M-alteratig path from a vertex i A 1, ad let A be all vertices i A that are reachable by a M-alteratig path from a vertex i B 1. Note that A 1 A = ad B 1 B = ad there are o edges of M joiig A ad B ; otherwise we have a bigger matchig tha M by makig a M-augmetig path, which is a cotradictio. Let A 3 = A (A 1 A ad B 3 = B (B 1 B. Let A i = a i ad B i = b i for i = 1,, 3. We cosider five cases to obtai lower bouds o κ(u, v depedig o the possible locatios of distict vertices u ad v. Case 1: u, v A, or u, v B. If u, v A, the sice every u, v-path must pass through a vertex i B, we have κ(u, v b = m + b 1. If u, v B, the we replace B ad b = m + b 1 i the proof of the case u, v A by A ad a = m + a 1. Case : u A ad v (A A, or u B ad v (B B. If u A ad v (A A, the sice every u, v-path must pass through at least oe vertex i B V (M, we have κ(u, v m. If u B ad v (B B, the we replace B V (M i the proof of the case u A ad v (A A by A V (M. Case 3: u A A ad v B B. Every u, v-path must pass through at least two vertices i M except the path of legth oe uv, which implies that κ(u, v m.
Thus, we have κ(g m( ( + b ( a1 + a b1 = m + a b ( 1( + a b1 m + (a 1 + b 1 ( b +a (. ( ( ( Sice o edge of M jois A to B, all vertices of A match ito B 3 uder M. Thus, we have a b 3. Similarly, we have b a 3. Thus, we have (a 1 + b 1 + (a + b ad (a + b m. Sice (a 1 + b 1 + (a + b 1 (a 1 + b 1 (a + b 1, we have (a 1 + b 1 (a + b 1 (. Thus, we have 8 κ(g m + (a 1 + b 1 ( b +a m + (a 1 + b 1 (a + b 1 (a + b ( 1 m + ( ( 8( 1 m = 9 8 m 3 8 8 m. To have equality i the last iequality of (, a 1 = 0, b = 0 or b 1 = 0, a = 0. We may assume that a 1 = 0 ad b = 0. To have equality i those iequalities a b 3 ad b a 3 below, a = b 3 ad b = a 3, which implies that a 3 = 0. Similarly, we have b 1 + a = ad b 1 = (a 1 from the iequalities (a 1 + b 1 + (a + b ad (a 1 + b 1 + (a + b 1 (a 1 + b 1 (a + b 1 below. Therefore, we have = a + b 1 + b = a + a + a = a. To have equality i the last iequality of (, every two vertices u ad v i B must have κ(u, v = m. Thus every vertex i B must be adjacet to all the vertices i A 1, which implies that oly the complete bipartite graph K a,3a satisfies the equality. Thus equality holds oly for K q,3q for a positive iteger q. 3 Average Edge-coectivity i Regular Graphs I this sectio, we itroduce a cocept i terms of edge-coectivity aalogously to average coectivity. A graph G is k-edge-coected if every subgraph obtaied by deletig fewer tha k edges is coected; the edge-coectivity of G, writte κ (G, is the maximum k such that G is k-edge-coected. The average edge-coectivity of a graph G with vertices, witte κ (G, is defied to be u,v V (G κ (u, v /(, where κ (u, v is the miimum umber of edges whose deletio makes v ureachable from u, which is same as the umber of edgedisjoit pathes betwee u, v. Note that κ (G κ (G = mi u,v V (G κ (u, v. This ew parameter shares certai properties with the average coectivity. Eve if we replace κ(g i Theorem.1 ad Theorem 3. by κ (G, the the iequalities hold. Theorem 3.1. If G has average degree d, ad V (G =, 5
Figure 1: A graph G 1 with κ(g 1 = 1 + O( q s ad κ (G 1 = q 1 d 1 κ (G d. The proof is the same as the proof of Dakelma ad Oellerma. Theorem 3.. For a coected graph G, κ (G α (G, (5 ad this is sharp oly for complete graphs with a odd umber of vertices. I additio, if G is a -vertex coected bipartite graph, the ( 9 κ (G 8 3 α (G, (6 8( 8 ad this is sharp oly for the complete bipartite graph K q,3q, where q is a positive iteger. The proof is the same as i Theorem 3. if we look at the secod vertex i sets of edge-disjoit paths. However, cosider the graph G 1 i figure 1, which is the graph obtaied from P s+1 by replacig a edge with a copy of K q. Two successive copies of K q share oe vertex. The total umber of vertices is 1 + s(q 1. Sice κ(g 1 = 1 + O( q ad s κ (G 1 = q 1, we have a big differece betwee the average coecitivity ad the average edge-coectivity of this graph. I order to aalyze average edge-coectivity of regualr graphs, we first itroduce a otio of i-balloo of a regular graph.if a regular graph G has a cut-edge, the we get compoets after we delete all cut-edges of G. We defie a i-balloo i G to be such a compoet icidet to i cut-edges. Note that a 1-balloo i G is a balloo i G itroduced i [9], where a balloo of a graph is a maximal -edge-coected subgraph, which is icidet to exactly oe cut-edge of G, ad for ay i 1, a i-balloo of a is a maximal -edge-coected subgraph of G except whe it is a sigle vertex, ad the resultig graph obtaied by shrikig each i-balloo to a sigle vertex is a tree. For a cubic graph, its smallest 1-balloo is the smallest possible balloo i a cubic graph, deoted by B 1 i [9]. The smallest -balloo i a cubic graph is K e. We deote the smallest i-balloo i a cubic graph by B i. Now we compute the average edge-coectivity of several cubic graphs with vertices less tha 10. Before doig it, we first give a lemma. 6
Lemma 3.3. If G has a vertex subset S i V (G such that [S, S] < δ(g, the S δ(g + 1. Furthermore, if G is a (r + 1-regular graph ad S is a vertex subset i V (G such that [S, S] < r + 1,the [S, S] S mod Proof. If S δ(g, the [S, S] S (δ(g ( S 1 δ(g. Thus the first statemet is true. For the secod statemet, (r + 1 S [S, S] is eve by the Degree-Sum Formula. Thus, we have [S, S] (r + 1 S S mod. By the Degree-Sum Formula, a odd regular graph has eve umber of vertices. If =, the it is K. Sice the edge-coecitivy of K is 3, we have κ (K ( ( = 3, which is smaller tha ( + 7 +58. If = 6, the κ (G = 3 sice if its edge-coectivity is less tha 3, the it has to have at least 8 vertices by Lemma 3.3. Hece the average edge-coectivity of a cubic graph with 6 vertices, 3, is greater tha frac ( ( 6 + 7 6+58 6. If = 8, the its edge-coectivity is at least sice if its edge-coecitivyt is equal to 1, the it has to have at least 10 vertices by Lemma 3.3. If its edge-coectivity is equal to, the it is the graph obtaied by addig two edges betwee two B 1s. Its edge-coectivity is ( 8 +, ad ote that 3 ( ( 8 8 +. Thus the average edge-coectivity of a cubic graph with 8 vertices is greater tha ( 8 + 7 8+58. Now, we prove that every cubic graph other tha K satisfies the boud i the followig theorem. Theorem 3.. If G is a coected cubic graph G with vertices, which is ot K, the ( ( 7 + 58 κ (G +. Proof. Cosider a miimal couterexample G with vertices. Claim 1: The edge-coectivity of G is 1. If ot, the κ (G ( ( ( + 7+58 for 10. I the above, we showed that a cubic graph with vertices less tha 10 satisfies the boud whe it is ot K. claim : Every 1-balloo of G is B 1. If D 1 is a 1-balloo of G with D 1 B 1 ad V (D 1 = 5 + a, the Degree-Sum formula guaratees that a is a eve positive iteger, which imples that a. Let G be the graph obtaied from G by replacig D with B 1. Note that G is cubic. I additio, sice G also has a cut-edge, 10 a = V (G V (G. Sice G is larger graph tha G, which is ot K, by the hopothesis of G, we have κ (G ( a + 7 ( a ( a + 9. By the costructio of G ad the fact that B 1 is -edge-coected, ( ( ( ( a 5 5 + a κ (G = κ (G κ (B 1 5( a 5+κ (B 1 +(5+a( a 5 ( a + 7 ( 9 5 + a ( a + 6 5( a 5 + + (5 + a( a 5 7
= ( + a + a a = + 7 7a + 9 ( + 7 + a + 11a + 3 6 + (5 + a(5 + a 1 + a( a 5 > ( + 7 + 9 for a, which is a cotradictio to the assumptio that G is a couterexample. Claim 3: Every -balloo of G is B 1. If D is a -balloo of G with D B 1 ad D = +a, the the Degree-Sum formula guaratees that a is a eve positive iteger, which imples that a. Let G be the graph obtaied from G by replacig D with B 1 i order that G is a cubic graph. Thus, G has a vertices for a, ad by the hypothesis that G is a miimal couterexample, κ (G ( ( a a + 7 ( a + 9. By the costructio of G, ad the fact that B 1 is -edge-coected, we have ( ( a κ (G = κ (G ( a ( = + a + a a ( = κ (B 1 ( ( + a ( a +κ (B 1 ( + a + ( + a( a + 7 9 ( a + 13 ( a + + 7 7a + 9 + 7 + 9 + a + 7a + 8 +(+a( a 13 + a( a + ( + a( + a 1 > ( + 7 + 9 for a, which is a cotradictio to the assumptio that G is a couterexample. Claim : G has o k-balloos for k 3. Assume that G has a k-balloo for k 3. Sice G cotais B 1 as a iduced subgraph by Claim 1, choose a k-balloo D k for k 3 which is closest to B 1. D k is icidet to B 1s or B 1 by the choice of D k. If k, the V (D k k sice each vertex i V (D k is icidet to at most oe cut-edge. Thus, we ca assume that V (D k = k + a with a 0. Suppose that there are m B s betwee B 1 ad D k. Let G be the graph obtaied from G by deletig all B s betwee D k ad B 1, deletig B 1, ad replacig D k with C k 1 ad attachig each cut-edge except oe betwee D k ad B k to each vertex i C k 1. Note that G has a m 6 vertices. Thus, we have κ (G ( ( a m 6 a m 6 + 7 ( a m 6 + 9. The costructio of G guaratees that ( ( ( a m 6 k 1 κ (G = κ (G κ (C k 1 ( ( m + 5 + a + k k + a (k 1( a m k 5 + + (κ (D k 1 ( ( 5 +m(κ (B 1 + (κ (B 5 1 + (m + 5 + a + k( m a k 5 8
( a m 6 + 7 ( 9 k 1 ( a m 6 + (k 1( a m k ( ( m + 5 + a + k k + a + + + 7m + 16 + (m + 5 + a + k( m a k 5 ( = + 7 +1 a 9 ( a+19+k+ka = + 7 +9 +1 ((a 9 + 63 ( 16 +k+ka + 7 +9, which is a cotradictio to the hypothesis that G is a couterexample. Thus, we ca assume that k is equal to exactly 3. Let V (D 3 = a. Note that a 1. Assume that there are m B s betwee B 1 ad D 3. Let G be the graph obtaied from G by deletig all B s betwee D 3 ad B 1, deletig B 1, replacig D 3 with B, ad attachig each of two remaiig cutedges to vertices of B with degree. Note that G has a m 1 vertices. Sice G is smaller tha G, we have κ (G ( ( a m a a m 1 + 7 ( a m 1 + 9. By the costructio of G, we have ( ( ( a m 1 κ (G = κ (G κ (B (( a m 5 ( ( ( ( a + m + 5 a 5 + +(κ (D 1 +m(κ (B 1 +(κ (B 5 1 +(a+m+5( a m 5 ( a m 1 + 7 ( 9 a + m + 5 ( a m 1+ 13 ( a m 5+ ( +7m + 16 + (a + m + 5( a m 5 = + 7 + 1 a 9 a + 87 + k ( = + 7 + 1 (a 9 + 615 ( 3 + k > + 7 + 9 + ( a,which is a cotradictio to the assumptio that G is a couterexample. Therefore, G cotais o k-balloos for k 3. By the above claims, the miimal couter example should be a graph cosists of oly B ad B 5. After cotractig each k-balloo of G, we should get tree with maximum degree, which is a path. Thus G should be two B 5 o the edvertices ad B s are attached each other. But i that case, κ (G = ( + 7 + 9, which satisfy the propositio. Thus, it cotradicts the assumptio that G is a couterexample. Ad the iequality is sharp oly for mod by the above example. Figure describes a ifiite family of graphs for which equality holds i Theorem 3.. More geerally, we cojecture for (r + 1-regular graphs. 9
Figure : A graph for which equality i Theorem 3. holds Cojecture 3.5. If G is a coected (r + 1-regular graph G with vertices, the ( ( ( κ (G mi{, + (r (r + r 1 + r3 + r + r 8 }. (r + 1 r + 1 If the above cojecture holds, the it is sharp for ifiitely may. Let A = P 3 + rk ad let B = K r+ e. Note that A is the smallest possible 1-balloo ad B is the smallest possible -balloo i a (r + 1-regular graph. Cosider a path P with legth at least 1. Replace both ed-vertices of P by A ad the other vertices of P by B. We defie a (r + 1- chai to be the resultig graph. A (r + 1-chai is (r + 1-regular ad satisfies the equality i the Cojecture 3.5. Aother usage of the graphs A ad B is to fid the miimum matchig umber over -vertex (l-edge-coected k-regular graphs ad a relatioship betwee eigevalues ad matchig umber i regular graphs. (See [9], [10], [] Pefect Matchigs i Regular Graphs The graphs achievig equality i Theorem 3. are also helpful to fid a lower boud for the umber of perfect matchigs over -vertex coected cubic graphs. I 189, Peterse [1] proved that every cubic graph without cut-edges has a perfect matchig. Thus, it is atural to ask how may perfect matchigs a -edge-coected cubic graph must have. I 1970s, Lovász ad Plummer [7] cojectured that every -edge-coected cubic graph with vertices has at least expoetially may (i perfect matchigs. The first result o the cojecture is the followig theorem proved by Edmods, Lovász, ad Pulleyblak [6]. Theorem.1. (Edmods, Lovász ad Pulleyblak [6]; Naddef [8] If G is a -vertex coected cubic graph without cut-edges, the m p (G +, where m p(g deotes the umber of perfect matchigs i G. Recetly, the cojecture was proved by Esperet, Kardos, Kig, Král, ad Nori [5]. With the result, they also proved more more geerally that every (k 1-edge-coected k-regular graph with vertices has at least expoetially may perfect matchigs. A k-regular graph G with κ (G k 1 has to have a perfect matchig. If we weake the coditio (k 1-edge-coectedess to has a perfect matchig, the how may perfect matchigs must a k-regular graph have? I this sectio, we aswer this questio for k = 3. 10
We will use Plésik s Theorem, which states that if G is the graph obtaied from a (k 1-edge-coected k-regular multigraph G by deletig at most k 1 edges i G, the G has a perfect matchig. Lemma.. If B is a balloo with the eck v i a cubic graph, where the eck of a balloo is the vertex with degree. the there are at least two ear perfect matchigs ot usig v. Proof. Let x ad y be the two vertices adjacet to v i B. Let B be the resultig graph obtaied from B by addig a edge betwee x ad y after deletig the vertex v. Note that B is a cubic multigraph without cut-edges. By Plésik s Theorem, there are at least two perfect matchigs ot usig the added edge, which implies that there are at least two ear perfect matchigs i B. Theorem.3. Every -vertex coected cubic graph with a perfect matchig except for K has at least four perfect matchigs. I additio, equality holds for all 3-chais. Proof. Assume that G is a coected cubic graph with vertices other tha K. Note that 6. If G has o cut-edges, the by Theorem.1, G has at least four perfect matchigs. Now, assume that G has a cut-edge. Hece we have at least two balloos (See Lemma 3.3 i [9]. By Lemma., each balloo has at least two ear-perfect matchigs ot usig its eck. Sice there are are at least two balloos, we have at least four perfect matchigs. Cosider a 3-chai G. There are exactly two ear perfect matchigs ot usig the eck of each copy of B 1 i G, ad sice every perfect matchig i G has to use all cut-edges i G, we have oly oe choice i each copy of K e. Thus, we have exactly four perfect matchigs i G. We believe that if for k, G is a coected k-regular graph with vertices, the m p (G (i is at least expoetially may. I particular, we cojecture the followig for odd regular graphs. Cojecture.. If for r 1, G is a coected (r + 1-regular graph with vertices, the for some cotat c, m p (G c(r 1!! (r+3 r+. Note that the cojecture. is true whe r = 1. umber c is equal to. I fact, whe r = 1, the costat Ackowledgmet The authors are grateful to Douglas B. West for his helpful commets. 11
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