On some variations of extremal graph problems

Discussiones Mathematicae Graph Theory 17 (1997 ) 67–76 ON SOME VARIATIONS OF EXTREMAL GRAPH PROBLEMS Gabriel Semanišin1 Department of Geometry and Algebra Faculty of Science, P.J. Šafárik University Jesenná 5, 041 54 Košice, Slovak Republic e-mail: [email protected] Abstract A set P of graphs is termed hereditary property if and only if it contains all subgraphs of any graph G belonging to P. A graph is said to be maximal with respect to a hereditary property P (shortly P-maximal) whenever it belongs to P and none of its proper supergraphs of the same order has the property P. A graph is P-extremal if it has a the maximum number of edges among all P-maximal graphs of given order. The number of its edges is denoted by ex(n, P). If the number of edges of a P-maximal graph is minimum, then the graph is called P-saturated and its number of edges is denoted by sat(n, P). In this paper, we consider two famous problems of extremal graph theory. We shall translate them into the language of P-maximal graphs and utilize the properties of the lattice of all hereditary properties in order to establish some general bounds and particular results. Particularly, we shall investigate the behaviour of sat(n, P) and ex(n, P) in some interesting intervals of the mentioned lattice. Keywords: hereditary properties of graphs, maximal graphs, extremal graphs, saturated graphs. 1991 Mathematics Subject Classification: 05C15, 05C35. 1 Research supported in part by the Slovak VEGA Grant. 68 G. Semanišin 1. Hereditary Properties of Graphs All graphs considered in this paper are ordinary and finite. The nature of our considerations allows us to restrict our attention to the set I of all mutually nonisomorphic graphs. For the sake of brevity, we shall say ”a graph G contains a subgraph H” instead of ”a graph G contains a subgraph isomorphic to a graph H”. A nonempty subset P of I is called hereditary property, whenever it is closed under subgraphs. In other words, if G is any graph from P and H is its subgraph, then H also belongs to P. A hereditary property is named additive, whenever it is closed under disjoint union of graphs. In what follows we shall deal with the following examples of hereditary properties: O = {G ∈ I : G is totally disconnected}, Ok = {G ∈ I : each component of G has at most k + 1 vertices}, Dk = {G ∈ I : G is k-degenerate}, Tk = {G ∈ I : G contains no subgraph homeomorphic to Kk+2 or K⌊ k+3 ⌋,⌈ k+3 ⌉ }, 2 2 Ik = {G ∈ I : G does not contain Kk+2 }. Any hereditary property P can be uniquely determined either by the set of graphs not appearing in P (even as a subgraphs) or by the set of maximal admissible graphs (for details see e.g. [1]). More precisely, let us define the sets F(P) of minimal forbidden subgraphs and M(P) of P-maximal graphs in the following manner: F(P) = {F ∈ I \ P : any proper subgraph F ∗ of F belongs to P}, M(P) = ∞ [ M(n, P), n=1 M(n, P) = {G ∈ P : |V (G)| = n and G + e ∈ / P for any edge e ∈ E(G)}. In the next sections, we shall often need the following useful lemmas. Lemma 1. Let P1 , P2 be any hereditary properties. Then the following statements are mutually equivalent: 1. P1 ⊆ P2 ; 2. for each H ∈ F(P2 ) there exists H ′ ∈ F(P1 ) such that H ′ ⊆ H; 3. for any positive integer n and an arbitrary G ∈ M(n, P1 ) there is G′ ∈ M(n, P2 ) such that G ⊆ G′ . On Some Variations of Extremal Graph Problems 69 P roof. (1) ⇒ (2). Let H ∈ F(P2 ). Then H ∈ / P1 , and clearly H is not a subgraph of any G ∈ P1 . Hence, there exists H ′ ∈ F(P1 ) such that H ′ ⊆ H. (2) ⇒ (3). If G ∈ M(n, P1 ), then G does not possess any H ′ ∈ F(P1 ). Thus G does not contain any H ∈ F(P2 ) and therefore either G ∈ M(n, P2 ) or there exists G′ ∈ M(n, P2 ) such that G ⊆ G′ . (3) ⇒ (1). This implication follows immediately from the definitions. Lemma 2. Let P1 and P2 be any hereditary properties of graphs. If P1 ⊆ P2 , G ∈ M(n, P2 ) and G ∈ P1 , then G belongs to M(n, P1 ). P roof. If G ∈ M(n, P2 ), then for each edge e of the complement of G we have G + e ∈ / P2 . Hence, G + e ∈ / P1 for any edge e ∈ E(G). Then since G ∈ P1 , we get G ∈ M(n, P1 ). It is not so difficult to see that for any hereditary property P, which is distinct from I, there exists the number c(P) (called the completeness of P) defined as follows: c(P) = max{k : Kk+1 ∈ P}. Given an arbitrary property P, we define the chromatic number of P as the minimum of the chromatic numbers of forbidden subgraphs of P and we denote it by χ(P). It is clear, that for each additive hereditary property P the value χ(P) is at least two. The following results describe the structure of additive hereditary properties of graphs. Theorem 1 [1]. Let L be the set of all hereditary properties. Then (L, ⊆) is a complete and distributive lattice in which the join and the meet correspond to the set-union and the set-intersection, respectively. Theorem 2 [1]. For every nonnegative k, Lk = {P ∈ L|c(P) = k} is a complete distributive sublattice of (L, ⊆) with the least element Ok and the greatest element Ik . 2. Two Extremal Graph Problems Many problems in graph theory involve optimization. One of them could be formulated in the following way: for a graph of given order a certain type of subgraphs is prohibited, and one is to determine the maximum possible number of edges in such a graph. A problem of this type was first formulated by Turán and his original problem asked for the maximum number of edges 70 G. Semanišin in any graph of order n which does not contain the complete graph Kp (i.e., he was interested in the number ex(n, Ip−2 ), see [2], [3], [9], [10], [12], [13]). A general extremal problem, in our terminology, can be formulated as follows. Given a family F(P) of forbidden subgraphs, find the number ex(n, P) = max{|E(G)| : G ∈ M(n, P)}. The set of P-maximal graphs of order n with exactly ex(n, P) edges is denoted by Ex(n, P). The members of Ex(n, P) are called P-extremal graphs. It is natural to investigate also the ”opposite side“, and therefore we define the number sat(n, P) = min{|E(G)| : G ∈ M(n, P)}. By the symbol Sat(n, P) we shall denote the set of all P-maximal graphs on n vertices with sat(n, P) edges. These graphs are called P-saturated. From the definitions immediately follows Proposition 1. Let P, P1 and P2 be arbitrary hereditary properties and G ∈ M(n, P). Then 1. sat(n, P) ≤ |E(G)| ≤ ex(n, P); 2. if 1 ≤ n ≤ c(P) + 1, then sat(n, P) = ex(n, P) = 3. ex(n, P) ≤ ex(n + 1, P) for every n; ¡n¢ 2 ; 4. if P1 ⊆ P2 , then ex(n, P1 ) ≤ ex(n, P2 ) for every n; 5. ex(n, P1 ∪ P2 ) = max{ex(n, P1 ), ex(n, P2 )} for n ≥ 1; 6. ex(n, P1 ∩ P2 ) ≤ min{ex(n, P1 ), ex(n, P2 )} for n ≥ 1. In [14] examples are presented, which demonstrate that unlike the number ex(n, P), the behaviour of sat(n, P) is not monotone in general. The following theorems present some fundamental results of extremal graph theory. The symbol α(G) denotes the number of vertices in a maximum independent set of G. Theorem 3 [11]. If P is a hereditary property with chromatic number χ(P), then µ ¶Ã ! 1 n ex(n, P) = 1 − + o(n2 ). χ(P) − 1 2 On Some Variations of Extremal Graph Problems 71 Theorem 4 [14]. If P is a given hereditary property and n u = u(P) = min |V (F )| − α(F ) − 1 : F ∈ P n o d = d(P) = min |E(F ′ )| : F ′ ⊆ F ∈ F(P) is induced by a set S ∪ {x}, S ⊆ V (F ) is independent and |S| = |V (F )| − u − 1, o x ∈ V (F ) \ S , then à ! 1 u+1 sat(n, P) ≤ un + (d − 1)(n − u) − , 2 2 if n is large enough. One can observe that in the case when the structure of F(P) is not known, the evaluation of the bound for sat(n, P) is much more complicated as the evaluation of the bound for ex(n, P). As a matter of fact, we can present hom-properties of graphs which were studied from this point of view in [4]. For that reason, in Section 3 we shall try to obtain another type of bounds for sat(n, P). However, as a consequence of the previous two theorems, we immediately have Corollary 1. If P is a hereditary property of graphs and sat(n, P) = ex(n, P) for every positive n, then χ(P) = 2. 3. Intervals of Monotonicity In spite of the fact that sat(n, P) is not monotone, we can prove some inequalities and estimations using the properties of the lattice of all hereditary properties. It will be shown that the class of k-degenerate graphs plays a ¢ ¡ (see e.g. [5]). very important role since, sat(n, Dk ) = ex(n, Dk ) = kn − k+1 2 Lemma 3. Let P1 , P2 be any hereditary properties and let P1 ⊆ P2 . If sat(n, P2 ) = ex(n, P2 ), then sat(n, P1 ) ≤ sat(n, P2 ). P roof. If G ∈ M(n, P1 ) then, by Lemma 1, there exists a graph H ∈ M(n, P2 ) such that G ⊆ H. Hence, |E(G)| ≤ |E(H)|. Since ex(n, P2 ) = |E(H)| = sat(n, P2 ), we obtain |E(G)| ≤ sat(n, P2 ). Therefore, sat(n, P1 ) ≤ |E(G)| ≤ sat(n, P2 ). 72 G. Semanišin Theorem 5. If Ok ⊆ P ⊆ Dk , n ≥ k + 1, then sat(n, P) ≤ kn − P roof. As already pointed out, sat(n, Dk ) = ex(n, Dk ) = kn− ¡ ¢ by Lemma 3, we have sat(n, P) ≤ kn − k+1 2 . ¡k+1¢ 2 ¡k+1¢ 2 . . Hence, The following lemmas describe two other criteria of monotonicity in L. Lemma 4. Let P1 , P2 be any hereditary properties. Then sat(n, P1 ∪ P2 ) ≥ min{sat(n, P1 ), sat(n, P2 )}. P roof. It is not difficult to see that M(n, P1 ∪ P2 ) is a subset of M(n, P1 )∪M(n, P2 ). Thus, sat(n, P1 ∪P2 ) cannot be less than the minimum of sat(n, P2 ) and sat(n, P1 ). Lemma 5. Let P1 and P2 be any hereditary properties of graphs, P1 ⊆ P2 , and let G be a graph of order n. If G ∈ P1 and G is P2 -saturated, then sat(n, P1 ) ≤ sat(n, P2 ). P roof. Lemma 2 yields G ∈ M(n, P1 ). Hence, by an application of Statement (1) of Proposition 1, we get sat(n, P1 ) ≤ |E(G)| = sat(n, P2 ). Theorem 5 provides an upper bound for sat(n, P) for the first part of interval (Ok , Ik ) in Lk . The next theorem covers the rest of this interval. In order to prove it, we have to recall that in [6] it was proved that for any F ∈ F(Dk ) holds δ(F ) ≥ k + 1. Theorem 6. If Dk ⊆ P ⊆ Ik , n ≥ k + 1, then sat(n, P) ≤ kn − ¡k+1¢ 2 . P roof. Since c(P) = k, we observe that Kk+2 ∈ / P. Hence, by Lemma 1, there exist graphs F ∈ F(Dk ) and H ∈ F(P) such that F ⊆ H ⊆ Kk+2 . But, as it was mentioned above, δ(F ) ≥ k + 1 and therefore F = H = Kk+2 . In addition, according to the definition of F(P), no graph of F(P) is properly contained in Kk+2 , which implies |V (F )| ≥ k + 2 for any F ∈ F(P). Now, let us define the graph Gkn with the vertex set V (Gkn ) = {v1 , v2 , . . . vn } in the following way (the symbol N (u) stands for the neighbourhood of the vertex u): N (vi ) = {v1 , v2 , . . . , vi−1 , vi+1 , . . . vn }, i = 1, 2, . . . , k, N (vi ) = {v1 , v2 , . . . , vk }, i = k + 1, k + 2, . . . , n. The graph Gkn does not contain a subgraph isomorphic to Kk+2 , but it is easy to see that after adding any edge e ∈ E(Gkn ) a copy of Kk+2 must On Some Variations of Extremal Graph Problems 73 appear in Gkn + e. Hence, Gkn ∈ M(n, Ik ). Furthermore, Gkn ∈ Dk and then, applying Lemma 2, Gkn ∈ M(n, P). This¡ implies, using Lemma 5, ¢ . Proposition 1, that sat(n, P) ≤ |E(Gkn )| = kn − k+1 2 4. Some Estimations of sat(n, P) and ex(n, P) In the previous section we have established a bound for sat(n, P) in the part of the interval (Ok , Ik ) of the sublattice Lk . The following theorem presents the exact value of sat(n, P) in one specific case. By the invariant κ(P) we understand the minimum of the numbers κ(F ), the vertex-connectivity number of F , running over all graphs F from F(P). We shall use the fact, proved in [7], that for any P-maximal graph G the value κ(G) is at least κ(P) − 1. Theorem 7. Let P be a hereditary property and let D1 ⊆ P ⊆ I1 . If κ(P) ≥ 1, then sat(n, P) = n − 1. P roof. By Theorem 6, we have sat(n, P) ≤ n − 1. An application of the fact, that the minimum degree of a graph from F(D1 ) is 2, and Lemma 1 yields that any F ∈ F(P) has a subgraph isomorphic to Cn for some n ≥ 3 (the symbol Cn stands for the cycle on n vertices). We distinguish two cases. Case 1. Let κ(P) = 1. Suppose indirectly that sat(n, P) ≤ n − 2 for some n. Then there exists a graph G ∈ M(n, P) with at most n − 2 edges. It is easy to see that G is disconnected. Let us denote by G1 , G2 , . . . , Gs , s ≥ 2, the components of G and let ri = |V (Gi )| for i = 1, 2, . . . , s. Since P each Gi has at least ri − 1 edges and si=1 ri = n, it follows that at least two components of G, say G1 , G2 , are trees. Then after adding any edge e = {u, v}, u ∈ V (G1 ), v ∈ V (G2 ), some F ∈ F(P) must appear in G + e. Since κ(G) = 1, we obtain F ⊆ (G1 ∪ G2 ) + e. But (G1 ∪ G2 ) + e is a tree which contradicts the fact that F contains a cycle. Case 2. Let κ(P) ≥ 2. If G ∈ M(n, P) then G is connected. Hence G has at least n − 1 edges. Therefore sat(n, P) = n − 1. The set of k-degenerate graphs is one with sat(n, P) = ex(n, P). It is widely known that the properties T2 (to be an outerplanar graph) and T3 (to be a planar graph) are other examples of such properties. We show that such properties have an exceptional position in the lattice L of all hereditary properties. 74 G. Semanišin Lemma 6. Let P1 ⊆ P2 ⊆ P3 be any hereditary properties of graphs and let f : {1, 2, . . .} → {0, 1, . . .} be a mapping. If ex(n, P1 ) = ex(n, P3 ) = f (n), then sat(n, P2 ) ≤ f (n) and ex(n, P2 ) = f (n). P roof. By Statement (4) of Proposition 1, we have f (n) = ex(n, P1 ) ≤ ex(n, P2 ) ≤ ex(n, P3 ) = f (n), which implies that ex(n, P2 ) = f (n). Since sat(n, P2 ) ≤ ex(n, P2 ) the assertion sat(n, P2 ) ≤ f (n) is also valid. Theorem 8. If P is a hereditary property, T2 ⊆ P ⊆ D2 , then sat(n, P) ≤ 2n − 3 and ex(n, P) = 2n − 3 for n ≥ 3. P roof. The proof follows from the fact that T2 ⊆ D2 and the number of edges of all T2 -maximal and D2 -maximal graphs of order n ≥ 3 is exactly 2n − 3. Lemma 7. Let P1 and P2 be any hereditary properties of graphs and let f : {1, 2, . . .} → {0, 1, . . .} be a mapping. If sat(n, P1 ) = sat(n, P2 ) = ex(n, P1 ) = ex(n, P2 ) = f (n), then 1. sat(n, P1 ∪ P2 ) = ex(n, P1 ∪ P2 ) = f (n); 2. sat(n, P1 ∩ P2 ) ≤ f (n) and ex(n, P1 ∩ P2 ) ≤ f (n). Furthermore, if there exists a graph G ∈ M(n, P1 ) ∩ M(n, P2 ), then ex(n, P1 ∩ P2 ) = f (n). P roof. (1) From the fact M(n, P1 ∪ P2 ) ⊆ M(n, P1 ) ∪ M(n, P2 ) it follows that sat(n, P1 ∪ P2 ) = ex(n, P1 ∪ P2 ) = f (n). (2) By Proposition 1, we have ex(n, P1 ∩ P2 ) ≤ min{ex(n, P1 ), ex(n, P2 )} = f (n). Since sat(n, P1 ∩P2 ) ≤ ex(n, P1 ∩P2 ), we obtain the desired inequality. Moreover, if there exists a graph G ∈ M(n, P1 ) ∩ M(n, P2 ), then G ∈ M(n, P1 ∩ P2 ). Clearly, |E(G)| = f (n). It immediately follows that ex(n, P1 ∩ P2 ) = f (n). It is easy to see that T3 and D3 are incomparable in the lattice L. So we can examine the lattice interval between T3 ∩ D3 and T3 ∪ D3 . Lemma 8. If n is a positive integer, n ≥ 4, then 1. sat(n, T3 ∪ D3 ) = ex(n, T3 ∪ D3 ) = 3n − 6; 2. sat(n, T3 ∩ D3 ) ≤ 3n − 6 and ex(n, T3 ∩ D3 ) = 3n − 6. P roof. As ex(n, T3 ) = ex(n, D3 ) = 3n − 6 for n ≥ 4, we have, by Lemma 7, that sat(n, T3 ∪ D3 ) = ex(n, T3 ∪ D3 ) = 3n − 6 and sat(n, T3 ∩ D3 ) ≤ 3n − 6. It is easy to see that there exists a graph G with 3n − 6 edges which is planar and 3-degenerate. It means G ∈ M(n, D3 ) and simultaneously On Some Variations of Extremal Graph Problems 75 G ∈ M(n, T3 ). Hence, G ∈ M(n, D3 ∪T3 ) and G ∈ M(n, D3 ∩T3 ). Therefore, by Lemma 7, ex(n, T3 ∪ D3 ) = 3n − 6. The next theorem is an immediate consequence of the previous two lemmas. Theorem 9. Let P be a hereditary property such that T3 ∩D3 ⊆ P ⊆ T3 ∪D3 . Then ex(n, P) = 3n − 6 and sat(n, P) ≤ 3n − 6 for n ≥ 4. 5. Reducible Hereditary Properties A generalization of a colouring of graphs leads us to the concept of reducible hereditary properties. Given hereditary properties P1 , P2 , . . . , Pn , a vertex (P1 , P2 , . . . , Pn )partition of a graph G ∈ I is a partition (V1 , V2 , . . . , Vn ) of V (G) such that for each i = 1, 2, . . . , n the induced subgraph G[Vi ] has the property Pi . A property R = P1 ◦P2 ◦ · · · ◦Pn is defined as the set of all graphs having a vertex (P1 , P2 , . . . , Pn )-partition (for more details see [1], [8]). The structure of extremal graphs with respect to reducible hereditary property is described by the following lemma. Lemma 9. If a graph G belongs to Ex(n, P1 ◦P2 ), then for each (P1 , P2 )partition of V (G) into two disjoint sets V1 , V2 the following holds: the induced subgraph G[V1 ] is P1 -extremal, G[V2 ] is P2 -extremal and G = G[V1 ] + G[V2 ]. P roof. If G is P1 ◦P2 -extremal, then obviously for any (P1 , P2 )-partition of V (G) into V1 and V2 holds G = G[V1 ] + G[V2 ] (otherwise we can add at least one edge, which is a contradiction to the extremality of G). Furthermore, if the graph G[V1 ] is not P1 -extremal, then then there exists a graph G∗ ∈ P1 of the same order with greater number of edges as G[V1 ]. Clearly, G∗ + G[V2 ] ∈ P1 ◦P2 and moreover, |E(G∗ + G[V2 ])| > |E(G[V1 ] + G[V2 ])|, which is again a contradiction. Thereby G[V1 ] is P1 -extremal. Analogous arguments work for G[V2 ] and that is why G[V2 ] is a P2 -extremal graph. As in [7] it was shown that χ(P1 ◦P2 ) = χ(P1 ) + χ(P2 ) − 1, we immediately have Theorem 10. If R = P1 ◦P2 is a reducible hereditary property, then 1 ex(n, R) = 1 − χ(P1 ) + χ(P2 ) − 2 µ ¶Ã ! n + o(n2 ). 2 76 G. Semanišin References [1] M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: V.R. Kulli, ed., Advances in Graph Theory (Vishwa Intern. Publication, Gulbarga, 1991) 41–68. [2] P. Erdös, Some recent results on extremal problems in graph theory, Results in: P. Rosentstiehl, ed., Theory of Graphs (Gordon and Breach New York; Dunod Paris, 1967) 117–123; MR37#2634. [3] P. 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