Equitable partition of planar graphs
Introduction
All graphs in this paper are simple and finite. A -partition of a graph is a collection of induced subgraphs such that is a partition of . Such a -partition is equitable if for all . If there is no confusion, then we use to denote a -partition of . We write to denote the maximum degree of a graph .
In 1970, Hajnal and Szemerédi [9] proved a conjecture of Erdős, stating that every graph admits an equitable -partition into empty subgraphs, if . In 2008, Kierstead and Kostochka [10] found a short proof. In 2010, Kierstead, Kostochka, Mydlarz, and Szemerédi [12] designed a fast algorithm to find such an equitable -partition. The bound on in the Hajnal–Szemerédi Theorem is sharp because of complete graphs for instance. Thus, there have been many results in this field trying to obtain better lower bounds on the number of parts for special graph classes. Motivated by Brooks’ theorem, Chen, Lih, and Wu [5] conjectured that a connected graph admits an equitable -partition into empty graphs if and only if it is not , an odd cycle, or (for odd ). They proved this conjecture for and Kierstead and Kostochka [11] proved the conjecture for . For planar graphs, Zhang and Yap [20] proved this conjecture for , and Nakprasit [15] proved it for ; in other words, he proved that every planar graph has an equitable -partition into empty subgraphs if .
If we relax the condition on each part, then it is possible to reduce the number of parts significantly. For instance, Williams, Vandenbussche, and Yu [17] proved that for all , every planar graph of minimum degree at least and girth at least has an equitable -partition into graphs of maximum degree at most .
We will mostly focus on the degeneracy of graphs. A graph is -degenerate if every non-null subgraph has a vertex of degree at most . Note that a graph is -degenerate if it has no edges, and -degenerate if it is a forest. Kostochka, Nakprasit, and Pemmaraju [13] studied the existence of an equitable -partition of a -degenerate graph into -degenerate graphs.
Theorem 1.1 Kostochka, Nakprasit, and Pemmaraju [13] For and , every -degenerate graph has an equitable -partition into -degenerate subgraphs.
This implies that every -degenerate graph admits an equitable 3-partition into -degenerate subgraphs, an equitable 9-partition into -degenerate subgraphs, an equitable 27-partition into -degenerate subgraphs, and an equitable 81-partition into forests.
Now we restrict our attention to planar graphs. As planar graphs are -degenerate, every planar graph admits an equitable 81-partition into forests. How far can we reduce 81? Esperet, Lemoine, and Maffray [7] proved that can be improved to .
Theorem 1.2 Esperet, Lemoine, and Maffray [7] For all , every planar graph admits an equitable -partition into forests.
However it is not known whether is tight. Indeed, Esperet, Lemoine, and Maffray [7] proposed the following problem:
Problem 1.3 Esperet, Lemoine, and Maffray [7] Does every planar graph admit an equitable 3-partition into forests?
This problem still remains open and is known to have affirmative answers in the following cases:
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is -degenerate, by Theorem 1.1 (even if is non-planar),
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the girth of is at least , due to Wu, Zhang, and Li [18],
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no two cycles of length at most share vertices in , due to Zhang [19],
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has no triangles, and no two cycles of length are adjacent, due to Zhang [19],
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has an acyclic -coloring, due to Esperet, Lemoine, and Maffray [7].
By relaxing the condition further, we may ask the following question.
Problem 1.4 For each , what is the minimum integer such that for all integers , every planar graph admits an equitable -partition into -degenerate subgraphs?
It is easy to see that by considering for large , see Meyer [14]. Since every planar graph is -degenerate, for all . Theorem 1.2 implies that . Not every planar graph admits a (not necessarily equitable) -partition into forests, shown by Chartrand and Kronk [4]. Thus, .
Our first and second theorems prove that and .
Theorem 2.1 Every planar graph admits an equitable 2-partition into -degenerate graphs.
Theorem 2.2 Every planar graph admits an equitable 3-partition into -degenerate graphs.
Our third theorem shows a weaker variant of Problem 1.3.
Theorem 3.1 Every planar graph admits an equitable 3-partition into two forests and one graph.
The rest of this paper is organized as follows. In Section 2, we prove Theorem 2.1, Theorem 2.2, and moreover, show that every triangle-free planar graph admits an equitable 2-partition into -degenerate graphs. In Section 3, we prove Theorem 3.1 and illustrate some discussions towards Problem 1.3 and its relative problems.
Section snippets
Equitable partition into degenerate graphs
For a graph and disjoint sets , of vertices of , we denote by the number of edges between and . If or , then we simply write or for . For a vertex set and vertices and , let us write for the set and for the set .
Our first theorem shows that .
Theorem 2.1 Every planar graph admits an equitable 2-partition into -degenerate graphs.
Proof Let be an -vertex planar graph. We proceed by induction on . We may assume
Equitable partition into forests and graph
In this section, we aim to prove the following theorem.
Theorem 3.1 Every planar graph admits an equitable 3-partition into two forests and one graph.
An acyclic -coloring of a graph is a proper -coloring such that there is no cycle consisting of two colors. In other words, if a graph has an acyclic -coloring, then its vertex set can be partitioned into independent sets , , , such that induces a forest for all . Borodin proved the following theorem, initially conjectured by
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Supported by the National Research Foundation of Korea (NRF), South Korea grant funded by the Korea government (MSIT) (NRF-2018R1C1B6003786), and by INHA UNIVERSITY, Republic of Korea Research Grant.
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Supported by the Institute for Basic Science, Republic of Korea (IBS-R029-C1).