Toughness of recursively partitionable graphs

A simple graph $G=(V,E)$ on $n$ vertices is said to be recursively partitionable (RP) if $G \simeq K_1$, or if $G$ is connected and satisfies the following recursive property: for every integer partition $a_1, a_2, \dots, a_k$ of $n$, there is a partition $\{A_1, A_2, \dots, A_k\}$ of $V$ such that each $|A_i|=a_i$, and each induced subgraph $G[A_i]$ is RP ($1\leq i \leq k$). We show that if $S$ is a vertex cut of an RP graph $G$ with $|S|\geq 2$, then $G-S$ has at most $3|S|-1$ components. Moreover, this bound is sharp for $|S|=3$. We present two methods for constructing new RP graphs from old. We use these methods to show that for all positive integers $s$, there exist infinitely many RP graphs with an $s$-vertex cut whose removal leaves $2s+1$ components. Additionally, we prove a simple necessary condition for a graph to have an RP spanning tree, and we characterise a class of minimal 2-connected RP graphs.


Introduction
Let n be a positive integer.An integer partition of n is a list a 1 , . . ., a k of positive integers such that a 1 ≤ a 2 ≤ • • • ≤ a k and a 1 + • • • + a k = n.Let G = (V, E) be a graph of order n.An (a 1 , ..., a k )-partition of G is a partition {A 1 , . . ., A k } of V such that |A i | = a i for all i.We say the partition has connected parts if, for all i ∈ {1, . . ., k}, the induced subgraphs G[A i ] are connected.
In 1976, Györi and Lovász considered the problem of determining when a graph has an (a 1 , . . ., a k )-partition with connected parts and independently proved the following theorem.
Theorem 1 (Györi-Lovász [14,20]).Let G be a graph of order n and a 1 , . . ., a k an integer partition of n.If G is k-connected, then it has an (a 1 , . . ., a k )-partition with connected parts.
We say G is arbitrarily partitionable (or just AP) if, for every integer partition a 1 , . . ., a k of n, there exists an (a 1 , . . ., a k )-partition of V with connected parts.AP graphs were introduced in [1], and a polynomial time algorithm for determining whether a subdivision of K 1,3 is AP was provided.
The graph G is recursively partitionable (RP) if G ≃ K 1 , or G is connected and satisfies the following recursive property: for every integer partition a 1 , . . ., a k of n, there is an (a 1 , . . ., a k )-partition {A 1 , . . ., A k } of V such that each G[A i ] is RP.RP graphs were introduced in [6,7].
In [7], RP trees were characterised (among other results), and in [6], a class of RP unicyclic graphs was characterised.In both papers, the authors made heavy use of the following characterisation of RP graphs.Proposition 2. [6] An n-vertex graph G = (V, E) is RP if and only if it is connected, and: • for every partition a, b of n, there is an (a, b)-partition {A, B} of V such that both G[A] and G[B] are RP.
RP graphs were independently introduced (as "partition wonderful graphs") as a result of investigations into rainbow-cycle-free edge colorings (such as in [15]), by Peter Johnson, with the help of Paul Horn, at the MASAMU 2020 workshop.
These graphs arise naturally when considering rainbow-cycle-free edge colorings (which are of recent interest in their own right: [12,16,19].)A JL-coloring of an n-vertex graph is an edge coloring using exactly n − 1 colors that does not contain any rainbow cycles.These colorings are studied for K n in [10] and [13], K n,m in [18] and complete multipartite graphs in [17].
In [15], the authors introduced the following standard construction for creating a JL-coloring of a connected graph G: 1.If n > 1, find a partition V = {A, B} with connected parts, 2. color edges between A and B with a single color that will not be used again,

iterate (1) and (2) on G[A] and G[B].
This leads to the main result of [15]: Theorem 3. [15] Every JL-coloring is obtainable by an instance of the standard construction.
Corollary 4. [15] Every JL-coloring of a connected graph G = (V, E) is the restriction of a JL-coloring of the complete graph with vertex set V .
Combining Proposition 2, Theorem 3 and Corollary 4 yields the following observation of Johnson: Observation 5. A connected graph G = (V, E) of order n is RP if and only if every JL-coloring ϕ of K n can be restricted to a JL-coloring ϕ| E of a copy of G.
The rest of this paper is organised as follows.In Section 2, we define useful graphtheoretical tools and constructions that will be used throughout the paper.In Section 3, we list basic observations about the properties of AP and RP graphs.In Section 4, we introduce recursive constructions of RP graphs, which we later use to find infinite classes of RP graphs with a given toughness.It is easy to see that if a graph has an AP (RP) spanning tree, then it is AP (RP).In Section 5, we take a more detailed look at spanning subgraphs of RP graphs and provide a necessary condition for an RP graph to have a spanning tree homeomorphic to K 1,k .We also show that if an RP graph has an RP spanning tree, then for every S ⊆ V we have c(G − S) ≤ |S| + 2. In Section 6, we find lower bounds for the maximum possible values of c(G − S) for S ⊆ V in an RP graph G.In particular, we show that, for any s, there exists an infinite family of RP graphs with a s-vertex cut whose removal leaves 2s + 1 components.In Section 7, we show that there exists a finite set of minimal RP graphs for any given possible cut size |S| and c(G − S).In Section 8, we bound c(G − S) from above, by showing that in an RP graph G, for any S ⊆ V , we have c(G − S) ≤ 3|S| − 1, which shows that every RP graph is A perfect matching of a graph is a set M of edges that are all pairwise disjoint, such that every vertex is incident with an edge in M .A near-perfect matching is a set M of edges that are all pairwise disjoint, such that every vertex except for one is incident with an edge in M .A graph is (near) matchable if it has a (near) perfect matching. Let ) and adding to it all edges of the form uv, where u Let G = (V, E) be a graph.Denote by c(G) the number of components of G.In particular, for a connected graph For a positive real number r, we say G is r-tough if τ (G) ≥ r.

Graph constructions
In this section, we define graph constructions that we will use throughout the paper.See Figures 1 and 2 for examples.
Let a, b, c be positive integers.The tripode graph T (a, b, c) is the tree that has one degree 3 vertex, v, the removal of which leaves three paths having a, b and c vertices.

Elementary and known results
In this section, we list a number of useful literature results on AP and RP graphs.We make frequent use of these results and observations, particularly Lemma The following lemma, by Bondy and Chvatal [9] is a somewhat well-known variation of Ore's Hamiltonicity Theorem [22].
With Lemma 8 we easily prove the following.
Proposition 10.Let G be a graph with σ(G) ≥ 2k and order n.If n ≤ 2k + 1, then G is RP (and therefore AP), and this bound is sharp.
Proof.The graph G is RP since it is traceable (Observation 7 and Lemma 8).
To prove the bound is sharp, consider the complete bipartite graph K k,k+2 .This graph has σ = 2k and order 2k + 2. However K k,k+2 does not have a perfect matching, and thus by Observation 7, it is not RP.
In [21], Marczyk showed that the above result can be improved for AP graphs with the extra condition α(G) ≤ ⌈ n(G) 2 ⌉.Theorem 11. [21] Let G be a connected graph of order n.
For G to have a (near) perfect matching, it is clearly necessary that α(G) ≤ ⌈ n(G) 2 ⌉.For a large class of graphs, including complete multipartite graphs, this condition is also sufficient.We summarize these equivalences in Proposition 12.
Note that there is no possible forbidden subgraph characterisation of AP (RP) graphs.Given any graph G of order n, the graph K n + G is Hamiltonian, and thus AP (RP).

Theorem 13. [7]
A tree is RP if and only if it is either a path, the tripode T (2, 4, 6), or a tripode T (a, b, c), where (a, b, c) is one of the triples in Table 1: Tripodes and balloons are "universal" for RP graphs with connectivity 1 and 2, respectively, as the following observation from [5] shows.
[5] Let S be a vertex cut of a connected graph G, let C 1 , . . ., C k denote the components of G − S, and let c i denote n(C i ).
To discuss AP and RP graphs of arbitrary connectivity, we find it easiest to work with the semistars K b0 (b 1 , . . ., b k ), as they are also "universal".
Proof.Notice that G is a spanning subgraph of K s (c 1 , . . ., c k ) and apply Lemma 6.
Per Observations 15 and 16, the triples (a, b, c) in Table 1 for which T (a, b, c) is RP are also the triples for which K 1 (a, b, c) is RP.

New RP graphs from old
In this section, we present two operations for combining RP graphs to obtain new RP graphs: the well-known sequential join, and a "subgraph replacement" operation.These constructions, in tandem with Lemma 6, allow us to easily prove that many graphs encountered in the rest of the paper are RP.Of particular interest is the use of replacement graphs in Section 6 to construct RP graphs with large vertex cuts that leave many components.
There is a generalisation of the fact that paths are RP.In particular, the sequential join of RP graphs is RP.

. , H k be RP graphs, and let
Proof.Let n i be the order of H i , and n = n 1 + • • • + n k the order of G.We proceed by induction on n.
The base case n = 1 is trivial, as then G ≃ K 1 , which is RP.
Let n ≥ 2, assume the proposition is true for all positive integers less than n, and let G be an n-vertex sequential join of RP graphs H 1 , . . ., H k .It suffices to show that for any a ∈ [1, n − 1], there exists a partition of G into two RP graphs G[A] and G[B] such that G[A] has order a.To do this, we will pick the subgraph induced by the 'leftmost' a vertices of G in a manner that breaks apart at most one of the graphs H i .
Let m 0 = 0, and for all i ∈ [1, k], let m i = i j=1 n j .Denote by s the largest nonnegative integer such that a ≥ m s .Since the graph H s+1 is RP, it can be partitioned into two RP parts H s+1 [X] and H s+1 [Y ], such that |X| = a − m s .We can thus pick A consequence of Proposition 17 is that the suspension K 1 + G of an RP graph G is RP.
Corollary 18. Suppose t is a positive integer.If K a0 (a 1 , . . ., a k ) and K b0 (b 1 , . . ., b j ) are RP, then so is the graph which is RP by Proposition 17.
Proof.We use induction on the order of the replacement graph.Clearly every replacement graph of order at most 3 is RP.Suppose that every replacement graph of order n − 1 or less is RP, and let Since H is a replacement graph, K is RP by definition.Thus, there is a partition and K[Y ] = K y0 (y 1 , . . ., y k ) are RP.Note that x i +y i = b i , and that we may have x i = 0 (y i = 0) for some i. and By the induction hypothesis, both H[X ′ ] and H[Y ′ ] are RP, so H is RP.

RP spanning subgraphs
It is clear that every graph with an RP (AP) spanning tree is RP (AP).In [7], it was shown that an AP graph need not have an AP spanning tree.Using the sequential join, it is easy to construct RP graphs that do not have a spanning tripode T (a, b, c) (and thus do not have an RP spanning tree).For example, the graph is RP but has no spanning tripode.In this section, we give a necessary condition for a graph to have a spanning tree homeomorphic to K 1,k (k ∈ N).There are two cases to consider.In both cases, we count the number of components G i that each path P j intersects.
Case 1: v / ∈ S. Assume without loss of generality that v ∈ G 1 .Let ζ(P i ) = |{j ≥ 2 : V (G j )∩V (P i ) = ∅}| be the number of components (other than G 1 ) that contain a vertex of P i .Between any two vertices of P i that lie in different components of G − S, there must be a vertex of S. Therefore, for all i, we have Each of the components G 2 , G 3 , . . ., G c must intersect at least one path P i , so Since the paths P i are disjoint, we have Combining Inequalities 1, 2 and 3, we obtain the following inequality But this contradicts the fact that c − s ≥ k ≥ 2.
Case 2: v ∈ S Let η(P i ) = |{j : V (G j ) ∩ V (P i ) = ∅}| be the number of components that contain a vertex of P i .Between any two vertices of P i from different components of G − S, there must be a vertex of S, however, it is possible that the end vertices of P i are not in S.
Thus, for all i, the following inequality holds Since every component G i intersects at least one path P i : Since the paths P i are disjoint, and none contain the vertex v ∈ S, we have Putting Inequalities 4, 5 and 6 together, we obtain But this contradicts the fact that c ≥ s + k.
By Theorem 13, every RP tree on at least 3 vertices is either a path (subdivided K 1,2 ) or a subdivided K 1,3 .Further, every RP balloon is spanned by a subdivided K 1,k with k ≤ 5, per Theorem 14.Thus, we have the following corollary.

Bounding c(G − S) from below
Let G be an RP graph and S ⊆ V (G).Per Theorem 13 and Observation 15, if |S| = 1, then c(G − S) ≤ 3, and the infinite family of RP tripodes {T (1, 1, 2k)} k∈N all achieve this bound.Theorem 14 and Observation 15 show that if |S| = 2, then c(G − S) ≤ 5, and the RP balloons {B(1, 1, 2, 3, 2k)} k∈N achieve this bound [7].In this section, we bound the maximum possible value of c(G − S) from below.In particular, we show that for all s, there are infinitely many RP graphs with an s-vertex cut S such that c(G − S) = 2s + 1.Further, we prove that there exists an RP graph G with a cut S such that |S| = 3 and c(G − S) = 8.
Proof.We first prove that G k = K 2 (1, 1, 2, 6, k) is RP for all k ∈ {1, . . ., 10}.Note that n(G k ) = 12 + k.Thus, it suffices to prove that for all λ ∈ {1, . . ., Table 2-11 list all the (subgraphs induced by) partitions needed to show that G k is RP for k ≤ 10.All the subgraphs induced by the partitions are RP either by Lemma 22, or by the previous cases.For example, the |A| = 5 row of Table 2 shows how to partition V (G 1 ) = {A, B} so that |A| = 5 and G 1 [A], G 1 [B] are both RP (see Figure 3).
To prove G k is RP for k ≥ 11, we use induction.Let k ≥ 11, assume G k is RP for all j < k, and let λ be any integer in {1, . . ., ⌊ 12+k 2 ⌋}.Then we can partition V (G k ) into two parts {A, B} where |A| = λ by picking is RP by induction, completing the proof.
Corollary 24.For all s ≥ 1, there exists an infinite family G s of graphs such that each graph G in G s has a vertex cut S with |S| = s and c(G − S) = 2s + 1.
These graphs are all RP by Theorems 13 and 23.Define H s+2 (j) inductively by setting By Theorems 19 and 23, the graph H s+2 (j) is RP.It's clear that the graph H s (j) has a vertex cut S with |S| = s and c(G − S) = 2s + 1 (for example, see Figure 4).To complete the proof, we let G s = {H s (j)} j∈N .
Remark 27.For each pair (b 0 , k) of positive integers, there are finitely many minimal (b 0 , k) RP semistars.
A well known theorem of Tutte states that a graph G has a perfect matching if and only if for every vertex cut S of G, the graph G − S has at most |S| odd components [23].The next lemma shows that this necessary condition can be generalised to partitions with connected parts of any size.
If S is a finite set, then let |S| k denote the number j in {0, 1, . . ., k − 1} such that |S| ≡ j (mod k).If G is a graph, and S ⊆ V (G), then let The following result is given in [5].
Lemma 28.[5] Let G be a connected graph, S a vertex cut of G with |S| < c(G − S), and k ≥ 2 a positive integer.If G is AP, then We give a slight sharpening of this lemma.The proof is similar to the proof in [5], with care taken to track the remainder term |V (G)| k k−1 .Lemma 29.Let G be a connected graph, and S a subset of V (G).If G has a partition into connected parts T 1 , T 2 , . . ., T m such that We begin by considering the following subgraph G ′ of G: To complete the proof, it suffices to show that |S * | ≥ w k (G * , S * ).Each component of G * − S * is of the form T i − S * for some i < m, and each such T i has exactly k vertices.Thus, we have Further, each vertex of G * is in some T i , i < m.The T i has at least one vertex of S * and at most k − 1 vertices not in S * .Therefore,  In either case, we derive a contradiction, so G does not have such a subgraph H.

Bounding c(G − S) from above
In this section, we show that RP graphs are 1 3 -tough.We have seen that there exist RP graphs with a cut-vertex v such that c(G − v) = 3.However, as we show in Theorem 34, for cuts S of greater size in RP graphs, we must have c(G − S) < 3|S|, and this bound is sharp when |S| = 2 or |S| = 3.
We say that an RP graph G of order n is minimal with respect to S, if there is no (λ, n − λ)-partition, for any λ, of G into RP graphs G 1 and G 2 such that G 1 is a proper induced subgraph of any of the connected components of G − S. Corollary 35.Every RP graph is 1  3 -tough.

Further Questions
We mention a few open questions.
By the induction hypothesis, both G[A] and G[B] are RP, completing the proof.

Theorem 20 .
Let G be a graph, k ≥ 2 a positive integer and S a subset ofV (G).If c(G − S) ≥ |S| + k, then G does not have a spanning subdivision of K 1,k .Proof.Let c(G − S) = c and |S| = s.Let G 1 , G 2 , . . ., G c denote the components of G − S.Assume contrary to the theorem statement that there is a subdivision T of K 1,k spanning G, and that c − s ≥ k.Denote by v the vertex of T such that d T (v) = k, and let P 1 , P 2 , . . ., P k be the k maximal paths of T − v.

Corollary 21 .
If G = (V, E) contains an RP spanning tree, then every S ⊂ V satisfies c(G − S) ≤ |S| + 2. If G is spanned by an RP balloon, then every S ⊂ V satisfies c(G − S) ≤ |S| + 4.

1 3
-tough.Finally, in Section 9, we list a set of open questions.
For a positive integer k, let E k denote the empty graph with k vertices and no edges.If G is a graph, then n(G) is its order (number of vertices), m(G) its number of edges.Let α(G) denote its independence number (the order k of a maximum induced E k subgraph), and κ(G) its vertex connectivity (the minimum cardinality of a set of vertices whose removal disconnects G).Let σ(G) = min{d(u) + d(v) : u and v are non-adjacent vertices of G}.A graph is traceable if it has a spanning path (i.e., a Hamiltonian Path) and Hamiltonian if it has a spanning cycle (i.e., a Hamiltonian cycle).
Thus, there are limitations to how low the toughness of a sequential join of RP graphs can be.However, replacement graphs provide RP graphs with high connectivity and low toughness (see Corollary 24).
b0 (b 1 , . . ., b k ) is RP, and {G i } k i=0 is a set of RP graphs such that n(G i ) = b i .Then the graph H is RP, where