SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov


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2 Serdica Math. J. 30 (2004), SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices of the graph G adjacent to any vertex of M. If v 1,..., v r is a vertex sequence in G such that Γ(v 1,..., v r ) = and v i is a maximal degree vertex in Γ(v 1,..., v i 1 ), we prove that e(g) e(k(p 1,..., p r )) where K(p 1,...,p r ) is the complete rpartite graph with p i = Γ(v 1,...,v i 1 ) \ Γ(v i ). We consider only finite nonoriented graphs without loops and multiple edges. The vertex set and the edge set of a graph G will be denoted by V (G) and E(G), respectively. We call pclique of a graph G a set of p pairwise adjacent vertices. A set of vertices of a graph G is said to be independent, if every two of them are not adjacent. We shall use also the following notations: e(g) = E(G) the number of the edges of G; G[M] the subgraph of G induced by M, where M V (G); Γ G (M) the set of all vertices of G adjacent to any vertex of M; d G (v) = Γ G (v) the degree of a vertex v in G; 2000 Mathematics Subject Classification: 05C35. Key words: Maximal degree vertex, complete spartite graph, Turan s graph.
3 96 Nickolay Khadzhiivanov, Nedyalko Nenov K n and K n the complete and discrete nvertex graphs, respectively. Let G i = (V i,e i ) be graphs such that V i V j = for i j. We denote by G 1 + +G s the graph G = (V,E) with V = V 1 V s and E = E 1 E s E, where E consists of all couples {u,v}, u V i, v V j ; 1 i < j s. The graph K p1 + + K ps will be denoted by K(p 1,...,p s ) and will be called a complete spartite graph with partition classes V (K p1 ),...,V (K ps ). If p p s = n and p 1,...,p s are as equal as possible (in the sense that p i p j 1 for all pairs {i,j}, then K(p 1,...,p s ) is denoted by T s (n) and is called the spartite nvertex Turan s graph. Clearly, e(k(p 1,...,p r )) = {p i p j 1 i < j r}. If p 1 p 2 2, then K(p 1 1,p 2 +1,p 3,...,p r ) K(p 1,p 2,...,p r ) = p 1 p 2 1 > 0. This observation implies the following elementary proposition, we shall make use of later: Lemma. Let n and r be positive integers and r n. Then the inequality {pi p j 1 i < j r} e(t r (n)) holds for each rtuple (p 1,...,p r ) of nonnegative integers p i such that p p r = n. The equality occurs only when K(p 1,...,p r ) = T r (n). In our articles [6, 7] we introduced the following concept: Definition 1. Let G be a graph, v 1,...,v r V (G) and Γ i = Γ G (v 1,...,v i ), i = 1,2,...,r 1. The sequence v 1,...,v r is called αsequence, if the following conditions are satisfied: v 1 is a maximal degree vertex in G and for i 2, v i Γ i 1 and v i has a maximal degree in the graph G i 1 = G[Γ i 1 ]. In [6, 7] we proved the following Theorem 1. Let v 1,...,v r be an αsequence in the nvertex graph G and there is no (r + 1)clique containing all members of the sequence. Then e(g) e(k(n d 1,d 1 d 2,...,d r 2 d r 1,d r 1 )) where d 1 = d G (v 1 ) and d i = d Gi 1 (v i ), i = 2,...,r. The equality holds if and only if G = K(n d 1,d 1 d 2,...,d r 2 d r 1,d r 1 ). Later αsequences appear in [1, 2, 3, 4] under the name degreegreedy algorithm. In [6] we obtained the following corollary of Theorem 1: If G is an nvertex graph and e(g) e(t r (n)), then either G = T r (n) or each maximal (in the sense of inclusion) αsequence in G has length r + 1. Later this corollary is published in [1] and [2]. Some other results about αsequences and its generalizations are given in our papers [8, 9]. R. Faudree in [5] introduces the following modification of αsequences:
4 Sequences of vertices in graphs 97 Definition 2. The sequence of vertices v 1,...,v r in a graph G is called β sequence, if the following conditions are satisfied: v 1 is a maximal degree vertex in G, and, for i 2, v i Γ i 1 = Γ G (v 1,...,v i 1 ) and d G (v i ) = max{d G (v) v Γ i 1 }. Obviously, the set of vertices of each βsequence is a clique. In the discrete graph K n each βsequence consists of one vertex only. If G is not a discrete graph, then there are βsequences of length 2. Let for K(p 1,p 2,...,p r ) we have p 1 p 2 p r and v i V (K i ), i = 1,...,r. Then v 1,...,v r is a βsequence. If v 1,...,v s is a βsequence in a graph G, then it may be extended to a maximal in the sense of inclusion βsequence v 1,...,v s,...,v r. It is clear that this extension is not contained in an (r + 1)clique and in this case Γ(v 1,...,v r ) =. Definition 3. Let v 1,...,v r be a βsequence in a graph G and V 1 = V (G)\Γ(v 1 ), V 2 = Γ(v 1 )\Γ(v 2 ), V 3 = Γ(v 1,v 2 )\Γ(v 3 ),...,V r = Γ(v 1,...,v r 1 )\Γ(v r ). We shall call the sequence V 1,...,V r a stratification of G induced by the β sequence v 1,...,v r and V i will be called the ith stratum. The number of vertices in the ith stratum will be denoted by p i. Let us note that V i = Γ i 1 \Γ i, where by Γ 0 we understand V (G), and Γ i = Γ G (v 1,...,v i ), i = 1,...,r. Hence V i Γ i 1 and V i Γ i =, thus V i V j = for i j. In addition v i Γ i 1 \Γ(v i ) implies v i V i. From V i Γ(v i ) = it follows that the vertex v i is not adjacent to any vertex of V i. Therefore d(v i ) n p i. But d(v) d(v i ) for every vertex v V i. Consequently, (1) so that (2) It is clear that d(v) n p i, for each v V i. V = r V i Γ r, i=1 r p i + Γ r = n. i=1 Analogically to Theorem 1 we obtain the following Theorem 2. Let v 1,...,v r be a βsequence in an nvertex graph G, which is not contained in an (r + 1)clique. If V i is the ith stratum of the stratification induced by this sequence and p i = V i, then (3) e(g) e(k(p 1,...,p r ))
5 98 Nickolay Khadzhiivanov, Nedyalko Nenov and the equality occurs if and only if (4) d(v) = n p i for each v V i, i = 1,...,r. Proof. The assumption that {v 1,...,v r } is not contained in an (r + 1) clique implies that Γ r = Γ(v 1,...,v r ) = and by (2) we have r i=1 p i = n. On the other hand, 2e(G) = r d(v) = d(v) i=1 v V i and by (1) it follows that v V (G) (5) r 2e(G) p i (n p i ) = 2e(K(p 1,...,p r )). i=1 So, (3) is proved. We have an equality in (3) if and only if there is an equality in (5), i.e. when (4) is available. Theorem 2 is proved. By (1) it follows immediately Proposition. If v 1,...,v r is a βsequence in an nvertex graph G, which is not contained in an (r + 1)clique, then r d G (v i ) (r 1)n. i=1 Note that the equality in (3) is possible also when G K(p 1,...,p r ). Example. Let Π be the graph which is the 1skeleton of a triangular prism. Let [a 1,a 2,a 3 ] be the lower base, [b 1,b 2,b 3 ] the upper base and [a i,b i ] the sideedges. Then a 1,b 1 is a βsequence in the graph Π. There is no 3 clique containing both a 1 and b 1. The stratification induced by this sequence is V 1 = {a 1,b 2,b 3 }, V 2 = {b 1,a 2,a 3 }. Therefore p 1 = p 2 = 3. So we have e(π) = 9 and e(k(3,3)) = 9, but Π K(3,3). Now we shall prove that in case of equality in (3), we may state something stronger than (4), but under an additional assumption about the βsequence from Theorem 2.
6 Sequences of vertices in graphs 99 Definition 4. Let v 1 be a vertex of maximal degree in a graph G. The symbol l G (v 1 ) denotes the maximal length of βsequences with first member v 1. Theorem 3. Let v 1,...,v r be a βsequence such that r = l G (v 1 ). Then, in the notation of Theorem 2, if e(g) = e(k(p 1,...,p r )), then we have G = K(p 1,...,p r ). Proof. It is clear that the assumption r = l G (v 1 ) guarantees that in G there is no (r + 1)clique containing the members of the sequence v 1,...,v r, so that Γ r = and (3) is valid (c.f. Theorem 2). Then the equality in (3) implies (4), we shall make use of below. In order to prove Theorem 3, we shall proceed by induction on r. For r = 1 we have one stratum V 1 with V 1 = p 1 = n and by assumption e(g) = e(k(p 1 )) = 0. Consequently G is a discrete graph, i.e. G = K(p 1 ). Let r 2 and suppose the statement to be valid for r 1. We shall prove first that V r = Γ r 1 \ Γ r = Γ r 1 is an independent vertex set. Suppose the contrary and let u 1 and u 2 be two adjacent vertices of V r. We shall prove that v 1,...,v r 1,u 1,u 2 is a βsequence in G. It is clear that v 1,...,v r 1 is a βsequence in G. By u 1 V r = Γ r 1 and the fact that all vertices of V r have the same degree (= n p r ) it follows that u 1 has a maximal degree among the vertices of Γ r 1. Clearly, [u 1,u 2 ] E(G) and u 2 V r = Γ r 1 imply that u 2 Γ(v 1,...,v r 1,u 1 ), besides that, u 2 has a maximal degree in Γ(v 1,...,v r 1,u 1 ). Hence, v 1,...,v r 1,u 1,u 2 is a βsequence in G. This contradicts the equality l G (v 1 ) = r. So, V r is an independent vertex set. Let G = G[V 1 V r 1 ]. Since V r is an independent set and d(v) = n p r for each v V r, then G = G + K pr and e(g) = e(g ) + p r (n p r ). It follows from this equality and e(k(p 1,...,p r )) = e(k(p 1,...,p r 1 )) + p r (n p r ) that (6) e(g ) = e(k(p 1,...,p r 1 )). Obviously, if M V (G ), then Γ G (M) = Γ G (M) V r. Thus, if v V (G ), then d G (v) = d G (v) + p r. It is clear then, that v 1,...,v r 1 is a βsequence in G. Hence, s = l G (v 1 ) r 1. We shall prove that s = r 1. Suppose the contrary and let v 1,u 2,...,u s be a βsequence in G with s r. We shall show that v 1,u 2,...,u s,v r is a βsequence in G. First of all, v 1,u 2,...,u s is obviously a βsequence in G as well. From s = l G (v 1 ) it follows that Γ G (v 1,u 2,...,u s ) = and therefore Γ G (v 1,u 2,...,u s ) = V r. This implies that v r Γ(v 1,u 2,...,u s ) and has a maximal degree in G among
7 100 Nickolay Khadzhiivanov, Nedyalko Nenov the vertices of Γ(v 1,u 2,...,u s ). Consequently v 1,u 2,...,u s,v r is a βsequence in G. This contradicts the equality l G (v 1 ) = r, since the length of the last sequence is greater than r. So, l G (v 1 ) = r 1. Note that the ith stratum of the stratification induced by this sequence in G is identical with the ith stratum of the stratification of G induced by the βsequence v 1,...,v r, i = 1,2,...,r 1. Taking into account (6), from the induction hypothesis we deduce that G = K(p 1,...,p r 1 ) and since G = G + K pr, then G = K(p 1,...,p r ). Theorem 3 is proved. Corollary 1. Let G be an nvertex graph, v 1 be a maximal degree vertex and l G (v 1 ) = r. Then e(g) e(t r (n)) and the equality appears if and only if G = T r (n). Proof. Let v 1,...,v r be a βsequence and p i denote the number of vertices in the ith stratum of the stratification generated by this sequence. According to Theorem 2, e(g) e(k(p 1,...,p r )). Lemma 1 implies that e(k(p 1,...,p r )) e(t r (n)). From the above two inequalities we get e(g) e(t r (n)). Let now e(g) = e(t r (n)). Then e(g) = e(k(p 1,...,p r )) and e(k(p 1,...,p r )) = e(t r (n)). From the first equality, according to Theorem 3, it follows that G = K(p 1,...,p r ). The second equality, according to Lemma 1, implies K(p 1,...,p r ) = T r (n). In this way we conclude that G = T r (n). Corollary 1 is proved. Corollary 2 ([6]). Let G be a graph with n vertices and v 1 be a maximal degree vertex, which is not contained in an (r + 1)clique, r n. Then e(g) e(t r (n)) and the equality occurs only for the Turan s graph T r (n). Proof. Let l G (v 1 ) = s. Then in G there is a sclique containing v 1, so s r and therefore e(t s (n)) e(t r (n)), where the equality is available only when s = r (see Lemma 1). Corollary 1 implies e(g) e(t s (n)) and we have equality only for G = T s (n). The above two inequalities imply the desired inequality e(g) e(t r (n)). It is clear that we have equality in the last inequality if and only if s = r and G = T s (n). Corollary 2 is proved. Evidently, Corollary 2 is a generalization of Turan s Theorem ([10]). Let G be an nvertex graph without (r + 1) cliques. Then e(g) e(t r (n)) and equality occurs only for G = T r (n). Corollary 3. Let G be an nvertex graph and v 1,...,v m be a βsequence, which is not contained in an (r + 1)clique, r n. Then e(g) e(t r (n))
8 Sequences of vertices in graphs 101 and in case m = r, we have e(g) = e(t r (n)) only when all the strata of the stratification induced by the sequence have almost equal number of vertices and the vertex degrees of any stratum are equal to the number of vertices out of the stratum. Proof. We shall extend the βsequence v 1,...,v m to a βsequence v 1,..., v m,..., v q not contained in a (q+1)clique. Obviously, m q r. Let V 1,...,V q be the strata of the βsequence v 1,...,v q and p i = V i. It follows from Theorem 2 that (7) e(g) e(k(p 1,...,p q )) and we have equality in (7) if and only if d(v) = n p i for each v V i, i = 1,...,q. On the other hand, according to Lemma 1 we have (8) e(k(p 1,...,p q )) e(t q (n)) e(t r (n)) where equality occurs only when q = r and p i are almost equal. Thus e(g) e(t r (n)). Equality appears here only when there is an equality in (7) and (8). The equality in (7) implies that d(v) = n p i for any v V i and the equality in (8) occurs if and only if q = r and p i p j 1 for each i,j. The extremal case in the proposition follows immediately from the above reasoning. Corollary 3 is proved. REFERENCES [1] B. Bollobas. Turan s theorem and maximal degrees. J. Combin. Theory Ser. B 75 (1999), [2] B. Bollobas. Modern Graph Theory. Springer Verlag, New York, [3] B. Bollobas, A.Thomason. Random graphs of small order. Ann. Discrete Math. 28 (1985), [4] J. A. Bondy. Large dense neighborhoods and Turan s theorem. J. Combin. Theory Ser. B 34 (1983), [5] R. Faudree. Complete subgraphs with large degree sums. J. Graph Theory 16 (1992),
9 102 Nickolay Khadzhiivanov, Nedyalko Nenov [6] N. Khadzhiivanov, N. Nenov. Extremal problems for sgraphs and the theorem of Turan. Serdica Bulg. Math. Publ. 3 (1977), (in Russian). [7] N. Khadzhiivanov, N. Nenov. The maximum of the number of edges of a graph. C. R. Acad. Bulgare Sci. 29 (1976), (in Russian). [8] N. Khadzhiivanov, N. Nenov. psequences of graphs and some extremal properties of Turan s graphs. C. R. Acad. Bulgare Sci. 30 (1977), (in Russian). [9] N. Khadzhiivanov, N. Nenov. On generalized Turan s Theorem and its extension. Ann. Sof. Univ. Fac. Math. 77 (1983), (in Russian). [10] P. Turan. An extremal problem in graph theory. Mat. Fiz. Lapok 48 (1941), (in Hungarian). Faculty of Mathematics and Informatics St. Kl. Ohridski University of Sofia 5, Blvd. James Bourchier 1164, Sofia, Bulgaria Received November 4, 2003
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