The 4-girth-thickness of the complete multipartite graph

The $g$-girth-thickness $\theta(g,G)$ of a graph $G$ is the smallest number of planar subgraphs of girth at least $g$ whose union is $G$. In this paper, we calculate the $4$-girth-thickness $\theta(4,G)$ of the complete $m$-partite graph $G$ when each part has an even number of vertices.


Introduction
The thickness θ(G) of a graph G is the smallest number of planar subgraphs whose union is G. Equivalently, it is the smallest number of parts used in any edge partition of E(G) such that each set of edges in the same part induces a planar subgraph. This parameter was introduced by Tutte [20] in the 60s. The problem to calculate the thickness of a graph G is an NP-hard problem [16] and a few of exact results can be found in the literature, for example, if G is a complete graph [2,5,6], a hypercube [15], or a complete multipartite graph for some particular values [21,22]. Even for the complete bipartite graph there are only partial results [7,13]. Some generalizations of the thickness for complete graphs have been studied, for instance, the outerthickness θ o , defined similarly but with outerplanar instead of planar [12], the S-thickness θ S , considering the thickness on a surface S instead of the plane [4], and the k-degree-thickness θ k taking a restriction on the planar subgraphs: each planar subgraph has maximum degree at most k [9].
The thickness has applications in the design of circuits [1], in the Ringel's earth-moon problem [14], and to bound the achromatic numbers of planar graphs [3], etc. See the survey [17].
In [19], the author introduced the g-girth-thickness θ(g, G) of a graph G as the minimum number of planar subgraphs of girth at least g whose union is G, a generalization of the thickness owing to the fact that the g-girth-thickness is the usual thickness when g = 3 and also the arboricity number when g = ∞ because the girth of a graph is the size of its shortest cycle or ∞ if it is acyclic. See also [11].
2 Calculating θ(4, K n 1 ,n 2 ,...,n m ) Given a simple graph G, we define a new graph G G in the following way: If G has vertex set V = {w 1 , w 2 , . . . , w n }, the graph G G has as vertex set two copies of V , namely, Figure 1: An edge of G produces four edges in G G.
On the other hand, an acyclic graph of n vertices has at most n − 1 edges and a planar graph of n vertices and girth g < ∞ has at most g g−2 (n − 2) edges, see [8]. Therefore, a planar graph of n vertices and girth at least 4 has at most 2(n − 2) edges for n ≥ 4 and at most n − 1, otherwise. In consequence, the 4-girth-thickness θ(4, G) of a graph G is at least Lemma 2.1. If G is a tree of order n then G G is a bipartite planar graph of size 2(2n−2).
Proof. By induction over n. The basis is given in Figure 1 for n = 2. Now, take a tree G with n + 1 vertices. Since it has at least a leaf, we say, the vertex w 1 incident to w 2 then we delete w 1 from G and by induction hypothesis, H H is a bipartite planar of size 2(2n − 2) edges for H = G \ {w 1 }. Since H is connected, the vertex labeled w 2 has at least a neighbour, we say, the vertex labeled w 3 , then u 2 v 3 v 2 is a path in H H and the edge u 2 v 2 / ∈ E(H H). Add the paths u 2 v 1 v 2 and u 2 u 1 v 2 to H H such that both of them are "parallel" to u 2 v 3 v 2 and identify the vertices u 2 as a single vertex as well as the vertices v 2 . This proves that G G is planar. To verify that is bipartite, given a proper coloring of H H with two colors, we extend the coloring putting the same color of v 3 to v 1 and u 1 . Then the resulting coloring is proper. Due to the fact that we add four edges, H H has 2(2n − 2) + 4 = 2(2(n + 1) − 2) edges and the lemma follows. Now, we recall that the arboricity number or ∞-girth-thickness θ(∞, G) of a graph G equals (see [18] : H is an induced subgraph of G .
We have the following theorem. Proof. Since G G has 2n ≥ 4 vertices, 4e edges and it follows the lower bound e n − 1 ≤ θ(4, G G).
To verify the upper bound, take an acyclic edge partition {F 1 , F 2 , . . . , F θ(∞,G) } of E(G). Therefore, and F i is the induced subgraph of the edge set F i for all i ∈ {1, 2, . . . , θ(∞, G)}). Indeed, an edge x j y j ∈ E(G G) is in F i F i if and only if w j w j ∈ E(G) is in F i . By Lemma 2.1, the result follows.
We prove it in the following way. Without loss of generality, u is a vertex in a part of size n m .
Proof. We need to show that G = K n 1 ,n 2 ,...,nm K n 1 ,n 2 ,...,nm . Let (W 1 , W 2 , . . . , W m ) be an m-partition of K n 1 ,n 2 ,...,nm . The graph K n 1 ,n 2 ,...,nm K n 1 ,n 2 ,...,nm has the partition (U 1 ∪ V 1 , U 2 ∪ V 2 , . . . , U m ∪ V m ) where U i and V i are copies of W i for i ∈ {1, 2, . . . , m}. Take two vertices x i and y j in different parts, without loss of generality, U 1 ∪ V 1 and U 2 ∪ V 2 . If the vertex x i is in U 1 and y j is in U 2 then they are adjacent because w i w j is an edge of K n 1 ,n 2 ,...,nm is m-complete. Similarly for x i ∈ V 1 and y j ∈ V 2 . If x i is in U 1 and y j is in V 2 , then also they are adjacent because w i w j is an edge of K n 1 ,n 2 ,...,nm . By Corollary 2.3 and Lemma 2.4, the theorem follows.
Due to the fact that θ(4, G) = θ(3, G) = θ(G) for any triangle-free graph G, we obtain an alternative proof for the thickness of the complete bipartite graph K 2n 1 ,2n 2 that is given in [7]. Corollary 2.6. If G = K 2n 1 ,2n 2 is the complete bipartite graph then θ(G) = e n−1 where n = n 1 + n 2 and e = n 1 n 2 .