On the k-residue of disjoint unions of graphs with applications to k-independence

Transcription

1 On the -residue of disjoint unions of graphs with applications to -independence David Amos 1, Randy Davila, and Ryan Pepper 3 1 Department of Mathematics, Texas A&M University Computational and Applied Mathematics Department, Rice University 3 Department of Computer and Mathematical Sciences, University of Houston - Downtown October 30, 013 Abstract The -residue of a graph, introduced by Jelen in a 1999 paper, is a lower bound on the -independence number for every positive integer. This generalized earlier wor by Favaron, Mahéo, and Saclé, by Griggs and Kleitman, and also by Triesch, who all showed that the independence number of a graph is at least as large as its Havel-Haimi residue, defined by Fajtlowicz. We show here that, for every positive integer, the -residue of disjoint unions is at most the sum of the -residues of the connected components considered separately, and give applications of this lemma. Our main application is an improvement on Jelen s bound for connected graphs which have a maximum degree cut-vertex. We demonstrate constructively that, in some cases, our extensions give better approximations to the -independence number than all nown lower bounds including bounds of Hopins and Staton, Caro and Tuza, Favaron, Caro and Hansberg, as well as Jelen s -residue bound itself. Additionally, we apply this disjoint union lemma to prove a theorem for function graphs those graphs formed by connecting vertices from a graph and its copy according to a given function) while simultaneously giving, in this context, different classes of non-trivial examples for which our new results improve on the -residue, further motivating our first application of the lemma. 1 Introduction Let G = V, E) be a simple, finite graph with order n = ng) and size m = mg). A set I V is an independent set if the vertices in I are pairwise nonadjacent. The cardinality of a largest independent set in G is called the Corresponding author: pepperr@uhd.edu 1

2 independence number of G and is denoted αg). The problem of computing independence number is nown to be NP-hard [14, 5]. As such, significant research has been devoted to finding sharp upper and lower bounds in terms of easily computable invariants see, e.g., [1, 11, 17, 6, 3, 33]). A generalization of independent sets was made by Fin and Jacobson [1, 13] and later by Hopins and Staton []. Given a positive integer, a set I V is a -independent set if the subgraph induced by I has maximum degree at most 1. The -independence number of a graph G, denoted α G), is the cardinality of a largest -independent set. The computation of α was shown to be NP-complete by Jacobson and Peters [3], and a large body of literature exists on -independence and related invariants see [, 3, 4, 5, 7, 9, 10, 19,, 4, 9]). The residue of a graph, introduced by Fajtlowicz [8], was conjectured to be a lower bound on the independence number by his conjecture-maing computer program Graffiti. This conjecture was proven by Favaron, Mahéo, and Saclé [10], and later by Griggs and Kleitman [16] and Triesch [31]. Jelen generalized the notion of residue to -residue and showed that this is a lower bound for -independence number for every [4]. In this paper we investigate some properties of -residue, extending Jelen s result and giving improvements under certain conditions. Moreover, we present an infinite family of graphs for which our result is an improvement on all nown tractable lower bounds for -independence number. For the entire paper, is assumed to be a positive integer. The maximum degree of a graph G will be denoted = G). The union of two graphs G and H is the disconnected graph G H with vertex set V G) V H) and edge set EG) EH). For compactness we denote the union of r copies of G by rg. For any v V G), the graph G {v} is obtained by deleting v from V G) and removing any edges incident to v from EG). We denote the complete graph on n vertices by K n and the star on n vertices by S n this is equivalent to the complete bipartite graph K 1,n 1 ). Before proceeding, we survey some nown results on the -independence number and introduce a few necessary tools. Known results on -independence We mention five tractable lower bounds on the -independence number, proven by: Hopins and Staton [], Favaron [10], Caro and Tuza [4], Jelen [4], and a corollary to a recent result of Caro and Hansberg []. Favaron s result was improved upon by Caro and Tuza, while Caro and Tuza s result was improved upon by Jelen and then later by Caro and Hansberg. Consequently, we will review only the lower bounds of Hopins and Staton, Caro and Hansberg, and Jelen all three of which are mutually non-comparable, as will be seen in the subsequent examples. We follow the notation used by Jelen and refer to the bound of Hopins and Staton as HS, and the bound of Caro and Hansberg by CH.

3 Theorem 1 Hopins-Staton []). Let G be a graph with maximum degree. Then, n α G) HS G) := 1 +. Theorem Caro-Hansberg []). Let G be a graph with average degree d. Then 1, α G) CH G) := n + d. These two bounds are non-comparable. For graphs where the maximum degree is far enough from the average degree, CH improves upon HS. For example, CH 1 S n ) = n 3 and HS 1S n ) = 1, thus for n > 3, CH 1 > HS 1. On the other hand, for regular graphs, HS improves upon CH. For example, let G be a cubic graph on n vertices and let =. Then HS G) = n and CH G) = n 5, thus HS > CH for every n. In order to introduce Jelen s bound, we need a few concepts. The degree sequence of G, denoted DG), is the sequence that lists the degrees of the vertices of G, which we will always write in non-increasing order. When we are required to be explicit, we will write the degree sequence as the sequence of distinct degrees of G with the number of vertices realizing each degree in superscript. For example, the degree sequence of the n-vertex star can be written DS n ) = {n 1, 1 n 1) }. We say a sequence of nonnegative integers is graphic if it is the degree sequence of some simple graph. If D is the degree sequence of G, then G is said to realize D, or, alternatively, G is a realization of D. The next theorem is due to Havel [0] but is also attributed to Haimi [18]. Theorem 3 Havel [0]). A sequence D = {d 1, d,..., d n } of non-negative integers in non-increasing order is graphic if and only if d 1 n 1 and the sequence D = {d 1, d 3 1,..., d d1+1 1, d d1+,..., d n } is graphic. The sequence D, obtained by deleting d 1 from D and reducing the next d 1 elements by one, is called the Havel-Haimi derivative of D. Higher derivatives will be denoted D i). This gives a well nown characterization of graphic sequences. Theorem 4. A sequence D = {d 1, d,..., d n } of non-negative integers in nonincreasing order is graphic if and only if for some i {0, 1,..., n 1}, the sequence D i) consists of n i zeros. The process of taing successive Havel-Haimi derivatives until a sequence of zeros is reached will be called the Havel-Haimi process. The residue of a graphic sequence D, denoted RD), is the number of zeros in the final sequence 1 In [], the authors define a -independent set as one whose induced subgraph has degree at most, as opposed to at most 1, leading to a slightly different formulation of the lower bound in the theorem. 3

4 of the Havel-Haimi process on D. When the context is clear, we will abbreviate R DG) ) as simply RG). A useful tool in the study of residue is the idea of majorization. Let A = {a 1, a,..., a n } and B = {b 1, b,..., b n } be two sequences of order n with the same sum. Then, A is said to majorize B if, for each positive integer t n, the following is true: t a i t b i. If A majorizes B, we write A B. Favaron et al. proved the following lemma, which was independently discovered by Griggs and Kleitman. Lemma 5 Favaron, Mahéo and Saclé [11], Griggs and Kleitman [16]). Let D 1 and D be graphic sequences of the same order for which D 1 D. Then RD 1 ) RD ). A third proof of Graffiti s conjecture was discovered by Triesch in 1996, who introduced the elimination sequence [31]. The elimination sequence of a graphic sequence D, denoted ED), is the union of the sequence of integers eliminated during the Havel-Haimi process and the resulting sequence of zeros. 3 In 1999, yet a another proof appeared, due to Jelen [4]. Jelen was able to prove much more than Graffiti s conjecture by defining a generalization ) of the residue of a sequence D, called -residue and denoted R ED), based on Triesch s elimination sequence. Let D be a graphic sequence and let E = ED) be the elimination sequence of D. The -residue of any graph realizing D is given by the sum R E) = 1 1 i)f i E), 1) i=0 where f i E) is the frequency of i in E. Since each graph has a unique degree sequence, and each degree sequence ) has a unique elimination sequence, we sometimes write R E) as R DG) or even R G). Note that if = 1, then Equation 1) gives the number of zeros in E, i.e. R 1 D) = RD). Jelen s wor now generalizes the results of Favaron et al., Griggs and Kleitman, and Triesch with the following two theorems. Lemma 6 Jelen [4]). Let D 1 and D be graphic sequences such that D 1 D. Then R D 1 ) R D ). Theorem 7 Jelen [4]). For any graph G, α G) R G). Griggs and Kleitman refer to majorization as dominance. Favaron, Mahéo, and Saclé never name the concept, instead defining the relation A B for two sequences A and B of the same length in an equivalent manner to our definition of A B. 3 The authors acnowledge the potentially ambiguous use of E to represent both the edge set of a graph and the elimination sequence of a degree sequence. In practice, we believe the meaning will be determined unambiguously by the context. 4

5 Jelen shows in [4] that R and HS are non-comparable. In Example of section we give a family of graphs for which R > CH. However, there also exist graphs for which CH > R. For a particular example of the latter, consider any cubic graph G of order 100. Then R G) = 75 and CH G) = 40. Thus, as claimed, the three bounds are mutually non-comparable. In the concluding remars of [4], Jelen notes that the R bound has room for improvement. In this paper we investigate the additive behavior of -residue over the disjoint union of graphs and present ways to improve Theorem 7 for disconnected graphs and special inds of connected graphs. 3 Lemmas and Tools The proof of Theorem 3 presented in [7] and [34] relies on a method of altering the edges of a graph without changing the degree sequence. Let G be a graph with order n 4 and suppose u, v, x, y V G) are four different vertices such that uv, xy EG) and ux, vy EG). A -switch illustrated in the figure below and sometimes called a Ryser switch) is performed by removing the edges uv and xy and adding the edges ux and vy. u x u x v y v y a) Before b) After Figure 1: A -switch These -switches can be used to transform a graph G realizing a graphic sequence D into a possibly non-isomorphic) graph H with the same degree sequence as G such that H has a maximum degree vertex v adjacent to a set of vertices of the next H) highest degrees a proof of this is given in the textboos by Merris [7] and West [34]). Furthermore, graphs with such a maximum degree vertex have the smallest -residues among all graphs realizing a given degree sequence, as we show in the next lemma. Lemma 8. Let D = {d 1, d,..., d n } be a graphic sequence with maximum degree and Havel-Haimi derivative D. Let G be any graph realizing ) D and let v V G) be a maximum degree vertex. Then R D ) R G {v}. Proof. Clearly the degree sums of D and D G {v} ) are equal, since both sequences are reached by deleting d 1 from D and reducing terms by one. The reductions in D occurred among the next highest degrees, while this was not necessarily the case in D G {v} ). It follows that for any positive integer t n, the sum of the first t degrees of D G {v} ) is at least as much as the sum of the first t degrees of D. Thus DG {v}) D, by definition of majorization. Therefore, by Lemma 6, R D ) R G {v} ). 5

6 Next we show that for large enough, R is calculated by an explicit formula. Lemma 9 Pepper [8]). If, then R G) = n m. Proof. Let E be the elimination sequence of DG). We expand the summand in the definition of -residue as follows: R G) = 1 1 i)f i E) i=0 1 = f i E) 1 1 if i E). i=0 During the Havel-Haimi process, the entirety of the degree sequence is either eliminated or reduced to zero and, at each step, the amount eliminated is equal to the amount reduced. Consequently, since the sum of the degree sequence is m, the sum of the elimination sequence is m. We consider two cases: +1 and =. For the case when + 1, we have that, 1 f i E) = n i=0 and 1 if i E) = m. Together with the expansion of R G) from above, this implies, Similarly, when =, R G) = n m. R G) = 1 i=0 f i E) 1 1 if i E) = n f G)) 1 m f E)) = n m = n m. Therefore, equality holds in both cases and the theorem is proven. It is easy to see that the residue of any complete graph is 1, i.e. RK n ) = 1. The following lemma generalizes this fact. Lemma 10. For positive integers and n, R K n ) = { +1, n n nn 1), n + 1 6

7 Proof. When n+1, the result follows from Lemma 9, so we focus on the case when n. It is easily verified, by performing the Havel-Haimi process, that the elimination sequence of K n is E = {n 1, n,...,, 1, 0}, so f i E) = 1 for i = 0, 1,..., n 1. Applying the definition of -residue, we have R K n ) = 1 1 i)f i E) i=0 = 1 [ ] + 1) + ) = 1 [ + 1) ] = + 1. Now we are ready to present the main result of our paper. 4 The Disjoint Union Lemma Few examples are required to discover that residue does not exhibit the same general additivity over the unions of graphs as other invariants, such as independence number. One such example is the union of two copies of C 4 and = 1. Then R 1 C 4 C 4 ) = 3 but R 1 C 4 ) = 4. The following theorem is somewhat surprising in this regard. We refer to both this theorem and its immediate corollary as the Disjoint Union Lemma. Theorem 11 The Disjoint Union Lemma). For any two disjoint graphs G and H, R G H) R G) + R H). Proof. We handle the case that first. Let G and H be any two graphs. By Lemma 9, R G H) = ng H) mg H). Since ng H) = ng)+nh) and mg H) = mg) + mh), we have mg) + mh) R G H) = ng) + nh) [ = ng) mg) ] [ + nh) mh) ] = R G) + R H). Thus equality holds whenever. For the case when <, we proceed by induction on ng H). Notice that the result is easily verified for any two graphs whose orders sum to 3 or less. Assume the theorem is true for all disjoint graphs whose orders sum to less than p, where p 4. 7

8 Let G and H be disjoint graphs whose orders sum to p. We may assume without loss of generality that both G and H have at least two vertices since the result is easily verified otherwise. Moreover, we may assume that there is a vertex v of maximum degree in G H which is in G if not, relabel the graphs). Now, perform -switches to G, if needed, until v is adjacent to a set of vertices realizing the degrees d, d 3,..., d +1 of DG). Let G = G {v}. Note that DG ) = D G), and both sequences have order p 1. Now, the elimination sequence of G and G are identical except that f EG )) = f EG)) 1. However, since <, the f term is not included in the -residue formula for either EG) or EG ). Thus R G ) = R D G) ) = R G). Furthermore, R D G H) ) R DG H) ), by Lemma 8. Finally, applying the inductive hypothesis and the facts discussed above, we have the following chain of inequalities whenever <, R G H) = R D G H) ) R DG H) ) R DG ) ) + R H) R G) + R H). Theorem 11 is easily extended to the number of components of any disconnected graph. Corollary 1. For any disconnected graph G with p components G i, R G) p R G i ) Equality for Theorem 11 is satisfied whenever, as was shown in the proof of the theorem. However, if the frequencies of the degrees in the elimination sequence of the union of two graphs is equal to the sum of the frequencies in the elimination sequences of the two graphs, considered individually, then the definition of -residue guarantees that the -residue of the union is equal to the sum of the -residues of the individual graphs, for every. We state this more compactly in the following observation. By the union of two elimination sequences, we mean the sequence obtained by simply combining the two sequences and arranging the resulting sequence in non-increasing order. Observation 13. For two graphs G and H, if EG H) = EG) EH) then R G H) = R G) + R H) for every. Remar. Note that the converse of Observation 13 is not true. For example, consider the sequence D = {3, 6, 1 }, which has the disjoint realization of two 8

9 copies of a graph H obtained by adding a degree one vertex to any vertex of a C 4. It is easily verified that R 1 D) = R 1 H) = 4. However, ED) = {3 1,, 1 3, 0 4 }, while EH) = {3 1, 1, 0 }. That is, R 1 D) = R 1 H) but ED) EH) EH). The union of two complete graphs realizes Observation 13, as can be seen by performing the Havel-Haimi process. In light of Corollary 1, this can be extended to the following result, which shows that the Disjoint Union Lemma is satisfied with equality for degree sequences realizable by the union of complete graphs with the -residue function behaving analogously to a linear operator under these circumstances. Theorem 14. Let n 1, n,..., n p, c 1, c,..., c p be a positive integers. Then, for every, p ) p R c i DK ni ) = c i R DKni ) ). Theorem 14 will prove fruitful in simplifying calculations in the examples to follow. In the next section, we show that families of graphs exist for which the difference between the sum of the residues of the components of a disjoint union and the -residue of the union can be arbitrarily large. These are precisely the graphs for which the Disjoint Union Lemma will offer the greatest improvement over Theorem 7 as a lower bound for α. 5 Applications of the Disjoint Union Lemma We now that R G)+R H) α G)+α H) by Jelen s result in Theorem 7. Moreover, since -independence number is additive componentwise, we have α G H) = α G) + α H). These facts, combined with The Disjoint Union Lemma, justify the next result. Theorem 15. For every positive integer and any disconnected graph G with p components G i, R G) p R G i ) p α G i ) = α G). Remar. Recall that the Disjoint Union Lemma is sharp for unions of complete graphs. The result is more interesting, with regards to Theorem 15, when the difference between the two sides of the inequality in the Disjoint Union Lemma is large. We will construct a family of graphs, each graph a disjoint union of two graphs, such that the sum of the -residues of the components grows arbitrarily larger than the -residue of the union, while staying relatively close to the 9

10 Figure : S 5 K 4 -independence number. Indeed, for the graphs in our family, the difference between the -independence number and the sum of the -residues is a function of only. Example 1. Let r and be positive integers. Consider the disjoint union of the star on r + 1 vertices and the corona of the complete graph on r vertices, denoted K r, and obtained by attaching a degree one vertex to each vertex of K r see Fig. ). First we compute R S r+1 ). Since DS r+1 ) = {r, 1 r }, then D S r+1 ) = {0 r } and ES r+1 ) = {r, 0 r }. We only consider the case that r, since otherwise equality holds for Theorem 11. Thus, whenever r R S r+1 ) = 1 1 i)f i ES r+1 )) = 1 r) = r. ) i=0 To compute R K r ) we note that D K r ) = {r r, 1 r }. process is as follows: The Havel-Haimi D K r ) = {r 1) r 1), 1 r 1), 0} D K r ) = {r ) r ), 1 r ), 0 } D K r ) = {r 3) r 3), 1 r 3), 0 3 }. D r) K r ) = {0 r }. Hence E K r ) = {r, r 1, r,...,, 1, 0 r }. Therefore, whenever r 10

11 R K r ) = 1 1 i)f i E K r )) i=0 = 1 [ ] r + 1) + ) = 1 [ 1) ] r + = r ) Finally, we compute R S r+1 K r ). Observe that By Theorem 14, DS r+1 K r ) = {r r+1), 1 r } = DK r+1 rk ). R S r+1 K r ) = R K r+1 rk ) = R K r+1 ) + rr K ). Since K ) = 1 and 1 r, Lemmas 9 and 10 give R K r+1 ) + rr K ) = + 1 Therefore, using equations ),3) and 4), R S r+1 ) + R K r ) R S r+1 K r ) = r + 1 = r + 1 = r ) r. 4) [ 1 ) r r + r + 1 which grows arbitrarily large as r. We now show that α S r+1 K r ) R S r+1 ) + R K r )) can be quite small and, in fact, depends only on. The -independence number of S r+1 is easily computed. ] α S r+1 ) = { r, 1 r r + 1, r + 1 5) For K r, if r + 1 then α K r ) = r. If 1 r, observe that taing each of the r degree one vertices and 1 vertices of degree r produces a largest -independent set. Hence, α K r ) = { r + 1, 1 r r r + 1 6) 11

12 As before, we only consider the case that 1 r, since equality for the Disjoint Union Lemma is satisfied otherwise. Applying equations ), 3), 5) and 6), observe now that the difference, α S r+1 K r ) R S r+1 ) + R K r ) ) = r + 1) r + 1 ) = 1 is a function of only. Note especially that, when = 1, the sum of the residues of the components is simultaneously equal to the independence number, and grows arbitrarily larger than the residue of the union as r. 5.1 The Disjoint Union Lemma and Connected Graphs In this section, we use the Disjoint Union Lemma to prove, for graphs with a maximum degree cut vertex, a theorem analogous to Theorem 11. Then we prove an upper bound for the -residue for a type of connected graph called a function graph Graphs with a maximum degree cut vertex Theorem 16. Let G be a graph with a maximum degree cut vertex c and let G 1, G,..., G p be the components of G {c}. Then for G), R G) p R G i ) p α G i ) α G). Proof. First we show that R G) p R G i ). Since G), R G) = R D G) ). It follows from Lemma 8 that, R G) = R D G) ) R G {c} ). Since D G {c} ) = D p G i), the Disjoint Union Lemma yields, ) p ) R G) R G {c} = R G i p R G i ). The second inequality in the theorem follows from Theorem 7. To see that the last inequality holds, let I be a largest -independent set of G {c}. Since I is also a -independent set in G, it follows that α G) p α G i ). Remar. To obtain the best lower bound for -independence number from Theorem 16, one should loo for a maximum degree cut vertex that maximizes the sum of the -residues of the components of G {c}. Additionally, we note that the conclusion of Theorem 16 does not hold for every cut vertex. For example, consider the graph in Fig. 3, we ll call it H. Removing c produces two copies of K 3. The elimination sequences of H and K 3 are easily computed: EH) = {3,, 1, 0 3 } and EK 3 ) = {, 1, 0 }. Note that R 1 H) R 1 K 3 ). 1

13 c Figure 3: The result of Theorem 16 does not hold for c. We now present a family of graphs for which Theorem 16 gives a significantly better lower bound for -independence number than Theorem 7. Example. Let, p, and r be positive integers such that p and r. Let G p,r be a graph obtained by taing p copies of Kr defined in Example 1) and joining the degree one vertices of each K r to a new vertex v see Fig. 4). Note that v is a cut vertex and that degv) = pr = G p,r ). Hence R G p,r ) pr K r ) by Theorem 16 above. Figure 4: G 3,4 Suppose p = 3t for some positive integer t and < r. Observe that, DG p,r ) = {pr, r pr, pr } D G p,r ) = {r 1) pr, pr } In D G p,r ), the term r 1) pr corresponds to the degree sequence of p copies of K r. Furthermore, since p = 3t for some positive integer t, the term pr = 3tr corresponds to the degree sequence of tr copies of K 3. That is, 13

14 D G p,r ) = DpK r trk 3 ). Since < r, R G p,r ) = R D G p,r )), applying Lemma 10, Theorem 14 and the equation above, we get, R G p,r ) = R pk r trk 3 ) = pr K r ) + trr K 3 ) + 1 ) = p + tr 3 3 ) = p r r ) Furthermore, pr K r ) = pr + 1 ), as was shown in Example 1. Thus we have, pr K r ) R G p,r ) = p r + 1 ) p r r ) r ) = p 1 which grows arbitrarily large as p or r. Thus the sum of the -residues of the components of G p,r {v} can be arbitrarily larger than the -residue of G p,r, giving an improvement on Theorem 7. In the next section, we show that for this family, Theorem 16 is a much better lower bound for -independence than both HS and CH Comparison of Theorem 16 to CH and HS To see that the Disjoint Union Lemma can, in some cases, yield better approximations than the lower bounds mentioned in the introduction, consider the family G p,r defined in Example. Note that, in terms of p and r, we have ng p,r ) = pr + 1, mg p,r ) = prr+3) and G p,r ) = pr. Hence the average degree is d = prr+3) pr+1. Let c be the maximum degree cut vertex of G p,r and let G 1, G,..., G j be the components of G p,r {c}. Recall from Theorem 16 that R G p,r ) j R G i ) α G p,r ). Moreover, taing a largest -independent set in each copy of ˆK r produces a largest independent set in G p,r since < r). Thus, from the results in Example, α G p,r ) = pr + 1) and p R G i ) = p r + 1 ). Choose p = r = and recall that p is a multiple of 3. We may express α G p,r ), j R G i ), R G p,r ), CH G p,r ) and HS G p,r ) as follows: 14

15 α G p,r ) = pr + 1) = p + p 3 p j R G i ) = pr + 1 ) = p + p 3 p + 1 ) R G p,r ) = p + pr 3 3 ) = p p p CH G p,r ) = n + d = pp + 1) 4p 3 p 1 + prr+3) pr+1 n HS G p,r ) = 1 + = p + 1 p p Here we use to mean asymptotically equal to. Observe that α G p,r ) j R G i ) R G p,r ). However, p R G i ) can be arbitrarily larger than any of the other three bounds Function Graphs We define a function graph, denoted G, f), as follows. Let G 1 and G be two copies of some connected graph G. Then G, f) is formed by adding edges to G 1 G according to a function f : V G 1 ) V G ). Function graphs were introduced and studied by Stephen Hedetniemi in [1]. More recently, the colorability and planarity of function graphs have been studied by Chen et al. in [6]; independence in function graphs is studied by Gera et al. in [15]. We will use the Disjoint Union Lemma to prove a bound on the -residue of a function graph G, f). Incidentally, the graphs G, f ), described in the proof below, give more non-trivial instances of graphs with a maximum degree cut vertex and for which Theorem 16 can be used to yield improvements over Jelen s -residue bound for -independence. Theorem 17. Let G, f) be a function graph. Then R G, f) R G) whenever G). Proof. Let u V G ) be a maximum degree vertex and let f be the constant function which maps each v V G 1 ) to u. First we will show that for any function f : V G 1 ) V G ), the following inequality is valid: R G, f) R G, f ). Let DG, f) = {d 1, d,..., d n } and DG, f ) = {d 1, d,..., d n}. Observe that n d i = n d i. Since, in G, f ) all n edges added by f are incident to u, which is not necessarily the case in G, f), observe also that for every t n, t d i t d i. It follows that DG, f ) DG, f), by definition of majorization, and hence R G, f ) R G, f), by Lemma 6. Now, R G, f ) = R D G, f ) ), since G). Perform -switches to G, if needed, until u is adjacent to the next G) vertices of highest degree of G. Delete u from G, f ) and observe that G, f ) {u} is the disjoint union of G 1 and G, where G is a realization of D G ). By Lemma 8, p 1 15

16 R D G, f ) ) R G, f ) {u} ) = R DG1 G ) ). From the Disjoint Union Lemma, R DG1 G ) ) R G 1 ) + R G ). Now, R G 1 ) = R G) and R G ) = R G), since G 1 is just a copy of G and G). It follows that R G 1 ) + R G ) = R G). Putting this all together, we have, R G, f) R G, f ) R G, f ) {u} ) = R DG1 G ) ) R G). 6 Concluding Remars We have seen how the Disjoint Union Lemma can give improvements on Theorem 7 for both disjoint unions and connected graphs with a maximum degree cut vertex. The examples given in this paper rely heavily on the result of Theorem 14; that is, that equality for the Disjoint Union Lemma is satisfied by degree sequences realizable by a disjoint union of complete graphs. This fact follows from Observation 13, which says that, for the union of two graphs A and B, equality holds for the Disjoint Union Lemma whenever EA B) = EA) EA). However, other inds of degree sequences exhibit this property, not just those from unions of complete graphs. For example, consider the degree sequence { 8 }, which has the disconnected realization C 3 C 5. A few simple calculations verify that EC 3 C 5 ) = { 3, 1, 0 3 }, EC 3 ) = {, 1, 0} and EC 5 ) = {, 1, 0 }. That is, EC 3 C 5 ) = EC 3 ) EC 5 ), but the degree sequence DC 3 C 5 ) = { 8 } is not realizable by the union of complete graphs. Notice that this is the same degree sequence as the example given in the first paragraph of section 4, and that R 1 D) = R 1 C 3 ) + R 1 C 5 ). One realization of this sequence gives equality for the Disjoint Union Lemma, while another does not. There are two problems, then, left unanswered. These are: 1) characterize the degree sequences with disconnected realizations that have the property described in Observation 13 and, ) more generally, characterize equality for the Disjoint Union Lemma. The example given in the remar after Observation 13 shows that these two problems, although related, are not equivalent to each other. Finally, we would lie to mention that Lemmas 5 and 6 actually show that residue and -residue are examples of a special ind of function from the theory of inequalities [30]. Given two sequences of real numbers A and B with order n, and a function f that maps those sequences to a real number, f is a Schur convex function provided that A B implies that fa) fb). Hence, both residue and -residue are Schur convex functions a fact that seems worth exploring further. References [1] Y. Caro, New results on the independence number, Tech. report, Tel-Aviv University,

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