THE THICKNESS OF THE COMPLETE MULTIPARTITE GRAPHS AND THE JOIN OF GRAPHS

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1 THE THICKNESS OF THE COMPLETE MULTIPARTITE GRAPHS AND THE JOIN OF GRAPHS YICHAO CHEN AND YAN YANG Abstract. The thickness of a graph is the minimum number of planar s- panning subgraphs into which the graph can be decomposed. It is known for relatively few classes of graphs, compared to other topological invariants, e.g., genus and crossing number. For the complete bipartite graphs, Beineke, Harary and Moon (On the thickness of the complete bipartite graph, Proc. Cambridge Philos. Soc., 60 (1964), 1 5.) gave the answer for most graphs in this family in In this paper, we derive formulas and bounds for the thickness of some complete k-partite graphs. And some properties for the thickness for the join of two graphs are also obtained. 1. Introduction A graph G is often denoted by G = (V (G), E(G)), where V (G) is the vertex set and E(G) is the edge set. The complement G of G is the graph whose vertex set is V (G) and whose edges are the pairs of nonadjacent vertices of G. A complete graph is a graph in which any two vertices are adjacent. A complete graph on n vertices is denoted by K n. The union G H of two graph G and H is the graph (V (G) V (H), E(G) E(H)). The join G + H of two vertex disjoint graphs G and H is obtained from G H by joining every vertex of G to every vertex of H. A complete k-partite graph is a graph whose vertex set can be partitioned into k parts, such that every edge has its ends in different parts and every two vertices in different parts are adjacent. K p1,p,...,p k denotes a complete k-partite graph in which the ith part contains p i (1 i k) vertices. And it is easy to see K p1,p,...,p k = K p1 + K p K pk. The thickness t(g) of a graph G is the minimum number of planar spanning subgraphs into which G can be decomposed. It was firstly defined by Tutte [7] in Since determining the thickness of a graph is NP-hard [4], it is very difficult to get the exact thickness number for arbitrary graphs. Beineke and Harary [] determined the thickness of the complete graph K n for n 4(mod 6) in 1965, while the remaining cases were solved in 1976, independently by Alekseev and Gončhakov [1] and by Vasak [8]. Theorem 1.1. [1,, 8] The thickness of the complete graph K n is t(k n ) = n+7 6, except that t(k 9 ) = t(k 10 ) = Mathematics Subject Classification. 05C10. Key words and phrases. thickness; complete multipartite graph; join. The work of the first author was supported by NNSFC under Grants No The work of the second author was supported by NNSFC under Grants No

2 YICHAO CHEN AND YAN YANG For the complete bipartite graphs, the problem has not been entirely solved yet. Beineke, Harary and Moon [3] gave the answer for most graphs in this family in Theorem 1.. [3] For m n, the thickness of the complete bipartite graph K m,n mn is t(k m,n ) =, except possibly when m and n are both odd and there (m+n ) exists an integer k satisfying m = k(m ) (m k). For the complete tripartite graph, Poranen proved t(k n,n,n ) n in [6] and Yang [9] gave the exact thickness number of K l,m,n (l m n) when l + m 5 and showed that t(k l,m,n ) = l+m when l + m is even and n > 1(l + m ) ; or l + m is odd and n > (l + m )(l + m 1). The reader is referred to [5] for more background and results about the thickness problems. In this paper, our concerning is the thickness of the complete multipartite graphs. Some results on the thickness for the join of two graphs are also given.. Thickness of some complete k-partite graphs The arboricity a(g) of a graph G is the minimum number of spanning forests into which the graph can be decomposed. It is known that the thickness of any graph is not less than a third of its arboricity and not more than its arboricity, and the arboricity of the complete graph K n is n. Lemma.1. Let K s + K n be the join of K s and K n (s, n 1), then t(k s + K n ) s. Proof. Denote the s vertices in K s by v 1,..., v s, and the n vertices in K n by u 1,..., u n. In the following, we will construct a planar subgraphs decomposition of K s + K n with s planar subgraphs, which shows t(ks + K n ) s. (1) Since a(k s ) = s, we have a spanning forests decomposition of Ks with s forests, and we denote these forests by F1,..., F s. () Add n parallel edges between v 1 and v in F 1 and insert vertices u 1,..., u n on these n parallel edges respectively. We will get a planar subgraph G 1 of K s + K n. (3) Add n parallel edges between v 3 and v 4 in F and insert vertices u 1,..., u n on these n parallel edges respectively. We will get a planar subgraph G of K s + K n. (4) Repeat this procedure with F i, for 3 i s. If s is odd, place the vertices u 1,..., u n in F s and join them to vs. We will get planar subgraphs G 3,..., G s of Ks + K n. For G 1 G G s = Ks +K n, a planar subgraphs decomposition of K s +K n with s planar subgraphs is obtained, and the lemma follows.

3 THE THICKNESS OF THE COMPLETE MULTIPARTITE GRAPHS AND THE JOIN OF GRAPHS3 Lemma.. For the complete k-partite graph K p1,p,...,p k number of odd numbers in the set {p 1, p,..., p k 1 }, then (k 3), let s be the pi s t(k p1,p,...,p k ) +. Proof. Suppose the k partite sets of K p1,p,...,p k are V 1 = {v 1 1, v 1,..., v 1 p 1 }, V = {v 1, v,..., v p },..., V k = {v k 1, v k,..., v k p k } respectively. We will construct one of its planar subgraphs decomposition as follows. (1) For vertices in V 1, join both v1 1 and v 1 to all the vertices in V i, 1 i k and i 1, we can get a planar graph G 1 1. Join both v3 1 and v4 1 to all the vertices in V i, 1 i k and i 1, we can get a planar graph G 1. Repeat this procedure with different vertices from V 1, until the p 1 th planar graph G 1 p 1 has been obtained. If p 1 is even, then all the vertices from V 1 will be used. If p 1 is odd, then the vertex vp 1 1 will not be used. () For vertices in V, join both v1 and v to all the vertices in V i, 1 i k and i, we can get a planar graph G 1. Join both v3 and v4 to all the vertices in V i, 1 i k and i, we can get a planar graph G. Repeat this procedure with different vertices from V, until the p th planar graph G has been obtained. p (3) Repeat this procedure with V 3,..., V k 1 respectively, we will get p p k 1 planar subgraphs of Kp1,p,...,p k, denote them by G 3 1,..., G 3 p3,..., G p k 1 1,..., G p k 1 p k 1 respectively. (4) Let G = G 1 1 G 1 p 1 Gp k 1 1 G p k 1 p k 1, then K p1,p,...,p k G = K s +K pk, in which s is the number of odd numbers in the set {p 1, p,..., p k 1 }. By Lemma.1, there exists a planar subgraphs decomposition of K s + K pk with s subgraphs. Above all, a planar decomposition of K p1,p,...,p k subgraphs is obtained, and the lemma follows. with pi s + planar Theorem.3. The thickness of the complete k-partite graph K p1,p,...,p k (k 3) k 1 ( p i equals when p i is even and p k > 1 k 1 p i ), or p i is odd ( k 1 ) ( k 1 ) and p k > p i 1 p i.

4 4 YICHAO CHEN AND YAN YANG ( k 1 k 1 Proof. When p i is even and p k > 1 p i ), or p i is odd and p k > ( ) ( k 1 ) p i 1 p i, from [3], the thickness of the complete bipartite graph K p1 +p +...+p k 1,p k is k 1 p i. Since K p1 +p +...+p k 1,p k is a subgraph of K p1,p,...,p k, by Lemma., we have pi s + t(k p1,p,...,p k ) t(k p1 +p + +p k 1,p k ) = k 1 in which s is the number of odd numbers in the set {p 1, p,..., p k 1 }. Since pi k 1 s p i + = where 0 s k 1, the theorem is obtained. 3. The thickness of G + K n Let p 1 = p = = p s = 1 and p s+1 = n, from Theorem.3, we have the following theorem. Theorem 3.1. The thickness of the join of K s and K n (s, n 1) equals s, if s is even and n > 1 (s ), or s is odd and n > (s 1)(s ). Theorem 3.. If G is a simple graph on p vertices, then the thickness of G + K n is p when p is even and n > 1 (p ), or p is odd and n > (p 1)(p ). Proof. Since K p,n G + K n K p + K n, we have t(k p,n ) t(g + K n ) t(k p + K n ). From Theorem 3.1 and the Theorem 1 in [3], the theorem can be obtained. Theorem 3.3. The thickness of the join of K s and K n (s 4, n 1) equals { 1, if s = 1, or s = 3 and n ; t(k s + K n ) =, if s = 3 and n 3, or s = 4 and n 1. Proof. It is easy to see that the graphs K 1 + K n, K + K n, K 3 + K 1 and K 3 + K are all planar graphs, so their thicknesses are all one. When n 3, the graph K 3 + K n contains a K 3,3 as its subgraph, so we have t(k 3 + K n ). On the other hand, it is trivial to construct a planar subgraphs decomposition of K 3 + K n with two planar subgraphs, using the construction of Lemma.1, which shows t(k 3 + K n ). Hence, t(k 3 + K n ) = when n 3. When n 1, the graph K 4 + K n contains a K 5 as its subgraph, so we have t(k 4 +K n ). And it is not hard to construct a planar subgraphs decomposition of K 4 + K n with two planar subgraphs, which shows t(k 4 + K n ). Hence, t(k 4 + K n ) = when n 1. p i,

5 THE THICKNESS OF THE COMPLETE MULTIPARTITE GRAPHS AND THE JOIN OF GRAPHS5 Summarizing the above, the theorem follows. The following result shows that the upper bound in Theorem 3.1 is best possible for s = 5. Corollary 3.4. When 1 n 1, the thickness of K 5 + K n is ; when n 13, the thickness of K 5 + K n is 3. Proof. From Theorem 3.1, the thickness of K 5 + K n is 3, when n 13. Because the graph K 5 +K n contains a K 5 as its subgraph, we have t(k 5 +K n ). Figure 1 constructs a planar subgraphs decomposition of K 5 + K 1 with two planar subgraphs, which shows t(k 5 + K n ), when 1 n 1. Summarizing the above, the corollary follows. f l 1 i b e k j h 5 a c d l g 3 4 d a g f j 1 5 i 3 h e b c 4 k Figure 1. A planar decomposition of K 5 + K 1 Though we do not find a formula of t(k s + K n ), for all s and n, we have the following upper and lower bounds. Corollary 3.5. For s, n 1 and s / {8, 9}, the possible thicknesses of K s + K n are consecutive integers in [ s+8, s 6 ]. Proof. Because t(k s + K n ) t(k s + K 1 ) and K s + K 1 = K s+1, combining it with Theorem 1.1, we have t(k s +K n ) s+8, except for s = 8 and 9. From Theorem 6 3.1, the maximum value of t(k s + K n ) equals s. Since the graph Ks + K n+1 is obtained from K s + K n by adding a new vertex and joining the new vertex to all vertices in K s, we have t(k s + K n+1 ) = t(k s + K n ) or t(k s + K n+1 ) = t(k s + K n ) + 1. Summarizing the above, the corollary is obtained. In a similar way, the following two corollaries can be obtained. Corollary 3.6. For n 1, the thickness of K 8 + K n is 3 or 4. Corollary 3.7. For n 1, the thickness of K 9 + K n is 3, 4 or 5.

6 6 YICHAO CHEN AND YAN YANG 4. The thickness of G + P n The graph P n is the path with n vertices. In this section we study the thickness of the join of a graph G and a path P n. Theorem 4.1. The thickness of the join of K s and P n (s, n 1) equals s, if s is even and n > 1 (s ), or s is odd and n > (s 1)(s ). Proof. In the proof of Lemma.1, we have constructed a planar subgraphs decomposition of K s + K n with s planar subgraphs by the procedure (1)-(4). By joining the vertices u 1, u,..., u n with a path in the process (), we can get a planar subgraphs decomposition of K s + P n with s planar subgraphs, which shows t(k s + P n ) s. On the other hand, Ks + K n is a subgraph of K s + P n, so we have t(k s + K n ) t(k s + P n ). Combining them with Theorem 3.1, the theorem follows. Theorem 4.. If G is a simple graph on s vertices, then the thickness of G + P n is s when s is even and n > 1 (s ), or s is odd and n > (s 1)(s ). Proof. With a similar proof to that of Theorem 3., the theorem can be obtained. Theorem 4.3. The thickness of the join of K s and P n (s 5, n 1) equals 1, if s = 1, or s = 3 and n = 1; t(k s + P n ) =, if s = 3 and n, or s = 4, or s = 5 and 1 n 1; 3, if s = 5 and n 13. Proof. It is easy to see that the graphs K 1 + P n, K + P n and K 3 + P 1 are all planar graphs, so their thicknesses are all one. When n, the graph K 3 + P n contains a K 5 as its subgraph, so we have t(k 3 + P n ). However, Figure illustrates a planar subgraphs decomposition of K 3 + P n with two planar subgraphs, which shows t(k 3 + P n ). So we have t(k 3 + P n ) = when n. u 1 u 1 u u u 3 v 1 v v 3 u 3 u n u n v 3 Figure. A planar decomposition of K 3 + P n

7 THE THICKNESS OF THE COMPLETE MULTIPARTITE GRAPHS AND THE JOIN OF GRAPHS7 When n 1, the graph K 4 + P n contains a K 5 as its subgraph, so we have t(k 4 + P n ). And Figure 3 presents a planar subgraphs decomposition of K 4 + P n with two planar subgraphs, which shows t(k 4 + P n ). So we have t(k 4 + P n ) = when n 1. u 1 u 1 u u 3 v 1 v u u 3 v 3 v 4 u n u n v 3 v 4 Figure 3. A planar decomposition of K 4 + P n From Theorem 4.1, the thickness of K 5 + P n is three, when n 13. Because the graph K 5 + P n contains a K 5 as its subgraph, we have t(k 5 + P n ). Figure 4 constructs a planar subgraphs decomposition of K 5 +P 1 with two planar subgraphs, which shows t(k 5 + P n ), when 1 n 1. Hence the thickness of K 5 + P n is two, when 1 n 1. Summarizing the above, the theorem follows. Proceeding as the proof of Corollary 3.5, we have the following corollaries. Corollary 4.4. For s, n 1 and s / {8, 9}, the possible thicknesses of K s + P n are consecutive integers in [ s+8, s 6 ]. Corollary 4.5. For n 1, the thickness of K 8 + P n is 3 or 4. Corollary 4.6. For n 1, the thickness of K 9 + P n is 3, 4 or The thickness for the join of two graphs G and H For arbitrary graphs G and H, we prove the following two properties for the thickness of the join of G and H. Property 5.1. Let G and H be two simple graphs with order p and q respectively, then t(g + H) Max{t(G), t(h)} + t(k p,q ). Proof. Because G + H is a subgraph of G H K p,q, in which the two partite sets of K p,q are V (G) and V (H) respectively, t(g + H) t(g H K p,q ) t(g H) + t(k p,q ). Since G and H are disjoint, t(g H) = Max{t(G), t(h)}, and the property follows.

8 8 YICHAO CHEN AND YAN YANG k h c a l 4 j 3 g 5 f 1 e b i d a b c d e f g h i j k l 3 Figure 4. A planar decomposition of K 5 + P 1 Property 5.. Let G and H be two simple graphs with order p and q respectively, then t(g + H) Min{t(H) + t(g + K q ), t(g) + t(h + K p )}. Proof. The graph G + H is a subgraph of H (G + K q ) and it is also a subgraph of G (H + K p ), in which V (K q ) = V (H) and V (K p ) = V (G). Hence both t(g+h) t(h)+t(g+k q ) and t(g+h) t(g)+t(h +K p ) hold, the property follows. References [1] V.B. Alekseev and V.S. Gončhakov, The thickness of an arbitrary complete graph, Math. Sbornik., 30() (1976), [] L.W. Beineke and F. Harary, The thickness of the complete graph, Canad. J. Math., 17 (1965), [3] L.W. Beineke, F. Harary and J.W. Moon, On the thickness of the complete bipartite graph, Proc. Cambridge Philos. Soc., 60 (1964), 1 5. [4] A. Mansfield, Determining the thickness of graphs is NP-hard, Math. Proc. Cambridge Philos. Soc., 93(9) (1983), 9 3.

9 THE THICKNESS OF THE COMPLETE MULTIPARTITE GRAPHS AND THE JOIN OF GRAPHS9 [5] P. Mutzel, T. Odenthal and M. Scharbrodt, The thickness of graphs: a survey, Graphs and Combin., 14 (1998), [6] T. Poranen, A simulated annealing algorithm for determining the thickness of a graph, Inform. Sci., 17 (005), [7] W.T. Tutte, The thickness of a graph, Indag. Math., 5 (1963), [8] J.M. Vasak, The thickness of the complete graph, Notices Amer. Math. Soc., 3 (1976), A-479. [9] Y. Yang, A note on the thickness of K l,m,n, Ars Combin., 117 (014), College of Mathematics and Econometrics, Hunan University, Changsha, China address: ycchen@hnu.edu.cn Department of Mathematics, Tianjin University, 30007, Tianjin, China address: yanyang@tju.edu.cn (Corresponding author: Yan YANG)