Ivančo, Jaroslav

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaJaroslav IvančoSome results on the Hadwiger numbers of graphsMathematica Slovaca, Vol. 38 (1988), No. 3, 221--226Persistent URL: http://dml.cz/dmlcz/130180Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1988Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath. Slovaca 38, 1988. No. 3, 221—226 SOME RESULTS ON THE HADWIGER NUMBERS OF GRAPHS JAROSLAV IVANCO 1. Introduction In the paper presented we prove some theorems generalizing the results in [4]. We also disprove two of Zelinka's conjectures. We consider only finite undirected graphs without loops and multiple edges. The concept of the Hadwiger number of a graph was defined in [2] (see also [4]), however, we use the equivalent definition introduced in [4]. Let G be a connected graph. Then a decomposition of its vertex set V(G) into nonempty subsets Vu ..., Vm with the following properties m (i) (J V^V{G), 1 = 1 (ii) Vfn Vj = 0 for any two integers i,j such that 1 ^ i ^ m, 1 ̂ j _̂  rn, i # j , (iii) Vt induces a connected subgraph of G for all i = 1, ..., m9 (iv) ViKjVj induces a connected subgraph of G for all i = 1, ..., m and all j= 1, ..., m, is called an //-decomposition of G. The Hadwiger number 77(G) of a connected graph is the maximal positive integer m such that there exists an //-decomposition of G into m subsets. Other definitions not given here will be found in [1]. 2. Bounds on 77(G) In this section we establish strict bounds of 77(G) depending on co(G) (i.e. the maximal number of vertices in a clique of G) and OQ(G) (i.e. the vertex covering number of G). Lemma 1. (B. Ze l inka [4]) Let G' be a subgraph of the graph G. Then 77(G') = 77(G). Theorem 1. Let G be a connected graph. Then 221 to(G) = n(G) = min(l + a0(G), [\(oy(G) + \V(G)\)]). Proof. As the Hadwiger number of the complete graph with n vertices is equal to w, the inequality co(G) = 17(G) follows evidently from Lemma 1. Now let us assume that Vu ..., Vn(G) is an //-decomposition of G. Let C be a minimal vertex covering set of G, i.e. |C| = a0(G). Then the set V(G) - C is an independent set of G. If 77(G) > 1 + «o(G) = 1 + |C|, then there exist at least two sets Vh \) (i i=- j) of the //-decomposition of G such that y.n c = VjnC = 0, i.e. VfuVj^ V(G) - C. So VtKj Vj is independent and it does not induce a connected subgraph of G, which is a contradiction. Thus 77(G) = 1 + 0,(0. (1) Let / b e a maximal subset of {1, ..., 17(G)} such that |J^| = 1 for iel. Then [J V{ induces a complete subgraph of G with |/| vertices and hence |/| ^ co(G). ief Since |^| = 2 forje{l, ..., 17(G)} - /, we get \V(G)\ = \VX\ + ... + \Vn(G)\ 2: |/| + 2(17(G) - |/|) :> 217(G) - co(G). This implies r1(G) = \(co(G) + \V(G)\). As l7(G) is an integer, we may write r1(G) = [\(co(G) + \V(G)\)]. (2) Thus (1) and (2) give the desired upper bound on 17(G). 3. The Hadwiger number of a complete multipartite graph The complete rc-partite graph is a graph whose vertices can be partitioned into k classes Uu ..., Uk such that two vertices are adjacent if and only if they belong to distinct classes. If \Ut\ = n{ for all i— 1, ..., k, then the complete k-partite graph is denoted by K(nx, ..., nk). B. Ze l inka [4] determined the Hadwiger number of a complete bipartite graph. Now we determine the Hadwiger number of a complete rc-partite graph for all k = 2. Theorem 2. Let k = 2,\=nx = ...=nkbe integers. For the complete k-partite graph K(nu ..., nk) there holds t](K(nu ..., nk)) = min(l +nx + ... + nk_l9 [\(k + nx + ... + nk)]). (3) 222 p r o o f For nk_x = 1 we have Oo(K(l, ..., 1, nk)) = k— 1 and co(K(l, ..., 1, nk)) - k and thus by Theorem 1 we get k ^ r)(K(\, ..., 1, nk)) = min(l + (k - 1), [\(k + (k - 1) + n,)]) = k. Hence 77(^(1, ..., 1, nk)) = min(l + (k - 1), [\(k + (k - 1) + n*)]). Now, let us assume that K(mx, ..., m )̂ (\ =mx = ... = mk) is a complete k-partite graph with the minimal possible number of vertices such that (3) does not hold. Then mk_x_\2, hence K(mx, ..., mk) — {w, v} (for vertices w, v belong-ing to classes of cardinalities mk _ x and mk respectively) is again a complete k-partite graph, i.e. K(mx, ..., mk_x — 1, mk— 1). This graph contains less vertices than K(m]9 ..., mk), therefore we may determine its Hadwiger number by (3). Two cases must be distinguished. Case 1. If k = 2 or k > 2 and mk> mk_ 2, then T](K(mx, ..., mk_x - 1, mk- 1)) = min(l +mx + ... + (mk_x - 1), [\(k + mx + ... +(mk_x - \) + (mk- l))]) = min(l + m1 + ... +mk_x, [i(k + mx + ...+mk)])-l. (4) Case 2. If k > 2 and mk = mk_2, then 7](K(mx, ..., mk_x -l,mk- 1)) = min(l +mx + ... +mk_3 + (mk_x - 1 ) + + (mk - 1), [\(k + mx + ... + mk_2 + (mk_x - 1) + (mk - 1))]) = = m i n ^ , + ... +mk_x, [\(k + mx + ... + mk)]) - 1. (5) However, in this case mk _ 2 = mk = mk _ x _\2 and hence m1 + . . .+m f c_ 2 + 2 + (mk_x + mk — 2mk) > k. This implies 2 + 2(mx + ... + mk) — 2mk > k + mx + ... + mk. Hence 1 +mx + ... +mk_x >\(k + mx + ... +mk) = [\(k + mx + ... + mk)]. As all numbers appearing in the relation are integers, we have mx + ... + mk_ x = [\(k + mx + ... + mk)]. (6) By (5) and (6) we get T](K(mu ..,, mh^\ -lfmk- l)) = min(l +mx + ... +mk_u [i(k + m. + . . . + m , ) ] ) - l . (7) 223 If VX9 ..., Vn is an //-decomposition of K(m]9 ..., mk) — {u9 v}9 then VJ, ..., V^, {w, t?} is evidently an //-decomposition of K(mX9 ..., mk). Thus ^ ( m , , ..., mk)) = 1 + v(K(mX9 ..., m*) - {w, i;}) = = 1 + 7](K(mX9 ..., mk_x -\9mk- 1)). So by (4) and (7) we get T](K(mX9 ...,m*)) = min(l + m, + ... +mk_x. [\(k + mx + ... + mk)]). (8) Since co(K(mX9 ..., mk)) = k and a0(K(m]9 ..., m*)) = m, + ... + m*_,, by Theorem 1 we have t](K(mX9 ..., mJ)_Smin(l + m, + ... +mk_X9 [\(k + m, + ... + m*)]). (9) (8) and (9) result in a contradiction to our assumption, which completes the proof. Remark 1. By Theorem 2 the complete multipartite graphs (graphs K(\9 ...9 ..., 1, n)) attain the upper (lower, respectively) bound in Theorem 1. Thus these inequalities cannot be improved in general. Remark 2. V. G. Vizing [2] suggested the study of the function Xk(n) which denotes the maximal possible number of edges of a graph with n vertices and with the Hadwiger number k. A. A. Zykov [5] and B. Ze l inka [3] proved that Xk(n) = (k - l )n - (*) for k _i 4, n = k. B. Ze l inka [3] con-jectured that this equality is true for any two positive integers n9 k9 where n^k. By Theorem 2 it is easy to prove that the complete m-partite graphs K(29 2, ..., ..., 2) (for m ^ 7) are counter-examples to Zelinka's conjecture. 4. The Hadwiger number of the Cartesian product of two stars The Cartesian product Gx x G2 of the graphs Gl9 G2 is the graph whose vertex set is the set of all ordered pairs [u]9 u2]9 where w, is a vertex of G, and u2 is a vertex of G2 and in which the vertices [uX9 u2]9 [vX9 v2] are adjacent if and only if either ux = vx and the vertices ul9 v2 are adjacent in G2, or u2 = v2 and the vertices M,, vx are adjacent in G,. B. Ze l inka [4] proved the following theorem: Let G,, G2 be two finite connected graphs, let GxxG2 be their Cartesian product. Then T / ( G , X G 2 ) ^ = *?(Gi) + 77(̂ 2) — 1. He also conjectured that the equality always holds in this relation. In this part we determine the Hadwiger number of the Cartesian product of two stars which will disprove Zelinka's conjecture. 224 First we prove Lemma 2. Let G and G' be connected homeomorphic graphs. Then tl(G) = tl(G'). Proof. Evidently, it is sufficient to consider only an elementary subdivi-sion. Hence let G be a connected graph and uv be its edge. Let G' be a graph which we obtain from G by deleting the edge uv and adding the vertex w and the edges uw and vw. Let Ul9 ..., Un{G1 be an //-decomposition of G' and without loss of generality let we Ux. If \UX\ = 1, then Ux — {w}9 Ul9 ..., Un{G1 is an //-decomposition of G. Thus 17(G) = //((?'). (10) If \UX\ = 1, then by (iv) rj(G') = 3. If T/(G') ^ 2, then J/(G) = 2 = //(G'), (11) because {w, t;} induces a complete subgraph of G. If r/(G') = 3, then G' contains a cycle, because the Hadwiger number of any tree is equal to 2. Since a subdivision does not give rise to a new cycle, the graph G must contain a cycle C. By Lemma 1 17(G) = 17(C) = 3 = J](G')9 (12) because the Hadwiger number of any cycle is equal to 3. Let VX9 ..., Vn{G) be an //-decomposition of G and without loss of generality let ve Vx. Then Vx u {w}9 Vl9 ..., Vn{G) is an //-decomposition of G' and so r/(G') = 17(G). (13) The assertion follows already from (10), (11), (12) and (13). Theorem 3. Let KXn and Kx m be stars, let Kx nxKx m be their Cartesian product. Then r](KhnxKhm) = 2 + min(n9 m). Proof. It can be easily seen that the graphs Kx nx KXm and K(l9 n9 m) are homeomorphic. By Lemma 2 we have i7(-«:, f l lx^ l tJ=i7(^(l ,H,m)). By Theorem 2 we get the assertion of Theorem 3. R e m a r k 3. As t](KUr) = 2, the Cartesian product Kx nxKUm (for n = 2, m ^ 2) is a counter-example to Zelinka's conjecture. 225 REFERENCES [1] HARARY, F.: Graph Theory, Addison-Wesley, Reading, Mass., 1969. [2] BИЗИHГ, B. T.: Heкoтopыe нepeшeнныe зaдaчи в тeopии гpaфoв, Уcпexи мaт. нayк, 23, 1968, 117—134. [3] ZELINKA, B.: On some graph-theoretical problems of V. G. Vizing, Čas. péstov. matem., 98, 1973, 56—66. [4] ZELINKA, B.: Hadwiger numbers of finite graphs. Math. Slov., 26, 1976, No 1, 23—30. [5] ЗЫКOB, A. A.: O кoличecтвe peбep гpaфa c чиcлoм Xaдвигepa нa бoлee 4, Пpиклaднaя мaтeмaтикa и пpoгpaммиpoвaниe, вып. 7, Кишинeв, 1972, 52—55. Recieved November 14, 1986 Katedra geometrie a algebry Prírodovedecká fakulta UPJŠ Jesenná 5, 041 54 Košice НЕКОТОРЫЕ РЕЗУЛЬТАТЫ О ЧИСЛАХ ХАДВИГЕРА ГРАФОВ а̂̂ о81аV 1уапсо Резюме В статье исследовано число Хадвигера графа. Показаны некоторые оценки для числа Хадвигера графа. Также определены числа Хадвигера полного ,-с-дольного графа и декартова произведения двух звезд. На основании этих результатов опровергнуты две гипотезы, кото­рые высказал Зелинка [3, 4]. 226 

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Some results on the Hadwiger numbers of graphs

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Institute of Mathematics AS CR, v. v. i.

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