On the maximum degree of path-pairable planar graphs

A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any $n$-vertex path-pairable planar graph must contain a vertex of degree linear in $n$.

not hold for k-path-pairable graphs. Observe that the stars S 2k (k ≥ 1) are k-path-pairable and yet have very low edge density and edge connectivity. On the other hand, a result of Bollobás and Thomason [1] shows that a 2k-connected graph with a lower bound on the edge density implies that G is k-linked. A similar theorem of Hirata, Kubota and Saito [8] states that a (2k + 1)-edge connected graph is (k + 2)-weakly-linked for k ≥ 2.
A k-path-pairable graph on 2k vertices is simply said to be path-pairable. Some of the most central questions in the study of path-pairable graphs concern determining the behaviour of their maximum degree. It is fairly easy to construct path-pairable graphs on n vertices (n even) with maximum degree linear in n. For example, complete graphs K 2n and complete bipartite graphs K m,n are path-pairable for all choices of m, n ∈ N with m + n even, m = 2, n = 2.
It is slightly more challenging to construct an infinite family of path-pairable graphs where the maximum degree grows sublinearly. We shall now describe such a family. Let K t be the complete graph on t vertices and let K q t be constructed from K t by attaching q − 1 leaves to the original vertices of K t . This family was introduced by Csaba, Faudree, Gyárfás, and Lehel [2], who also proved that K q t is path-pairable as long as t · q is even and q ≤ t 3+2 √ 2 . The bound on q has been recently improved to ≈ 1 3 t [7]. Observe that n = |V (K q t )| = t · q and ∆(K q t ) = t + q − 2 = O( √ n) when q = Ω(t). Additional path-pairable constructions with maximum degree c √ n can be found in [9] and [11].
The following result due to Faudree [4], shows that the maximum degree of a pathpairable graph has to grow with the order of the graph.
Theorem 1.1. If G is path-pairable on n vertices with maximum degree ∆, then n ≤ 2∆ ∆ .
Letting ∆ min (n) := min{∆(G) : G is a path-pairable graph on n vertices}, this result is equivalent to ∆ min (n) ≥ c 1 log n log log n , for some constant c 1 . To date, the best known upper bound on ∆ min (n) is due to Győri, Mezei, and Mészáros, exhibiting a path-pairable graph with maximum degree ∆ ≈ 5.5 · log n [6]. In summary, we have the following general asymptotic bounds on ∆ min (n): We are interested in determining the behaviour of the maximum degree in path-pairable planar graphs. Let us define ∆ p min (n) to be min{∆(G) : G is a path-pairable planar graph on n vertices}.
Faudree, Gyárfás and Lehel [5] proved that a path-pairable graph without a K 1,n must have at least 3n/2 − log n − c edges, for some absolute constant c. Although planar graphs might have considerably more than (3/2 − o(1))n edges we wished to determine whether planarity would be enough to force a vertex of linear degree.
We first note that a simple application of the Planar Separator Theorem of Lipton and Tarjan [10] shows that every path-pairable planar graph on n vertices must contain a vertex of degree at least c √ n. Indeed, if G is such a graph, then the Separator Theorem allows us to partition V (G) into three sets S, A, B, where |S| = O( √ n), |A| ≤ |B| ≤ 2n/3, and there are no edges between A and B. Now, while path-pairable graphs G need not be highly connected or edge connected, they must satisfy certain connectivity-like conditions. More precisely, they must satisfy the cut-condition: for every subset X ⊂ V (G) of size at most n/2, there are at least |X| edges between X and V (G) \ X. Note that the cut-condition is not sufficient to guarantee path-pairability; see [11] for additional details. Accordingly, since n/4 < |A| < n/2 and there are no edges between A and B, the cut-condition implies that there are at least |A| edges between A and S. We therefore obtain a vertex in S of degree Ω( √ n).
Our main theorem, which we state below, shows that we can do much better than this.
Namely, every path-pairable planar graph must have a vertex of linear degree.
Theorem 1.2. There exists c ≥ 10 −10 10 such that if G is a path-pairable planar graph on n vertices then ∆(G) ≥ cn.
We have not made an attempt to optimize the constant c obtained in the proof. The value we give is surely far from the truth.
However, there is currently a significant gap between the constants in the upper and lower bounds. Closing this gap and finding the truth is an interesting open problem.

Organization
The remainder of the paper is organized as follows. In the next short section, we shall describe our construction establishing Theorem 1.3. The third section of this paper contains a proof of our main theorem, Theorem 1.2. This proof relies on three preparatory lemmas and on some common facts concerning planar graphs. In particular, we use heavily the fact that any subset X of the vertices of a planar graph induces less than 3|X| edges, and any bipartite planar graph on n vertices has less than 2n edges. Finally, we close with some remarks and open problems.

Notation
Our notation is standard. Thus, for a graph G and two subsets X, Y ⊂ V (G) we say that denotes the graph induced in G with vertex set X.

The Construction
Our aim in this section is to prove Theorem 1.3, which we restate here for convenience. 2. If u ∈ {x AB , x BC , x CA } and v ∈ A∪B ∪C such that there exists a direct edge between u and v, join them by this edge. It is straightforward to check that the above instructions find edge-disjoint paths joining terminals, regardless of the choice of P.
3 The Proof of Theorem 1.2 The aim of this section is to prove our main theorem, Theorem 1.2. Our prove is based on three preparatory lemmas. First, we shall introduce some terminology. Let G be a multigraph. We say that two multiedges e, f of G are at distance d if the shortest path in G joining an endpoint of e and an endpoint of f has length d. If two multiedges are at distance 0, we shall simply say they are incident. Further, we shall refer to a matching of size k as a k-matching. We say that a k-matching is good if every pair of edges in the matching is at distance exactly 1. Notice that contracting all the edges of a good k-matching results in the complete graph K k (with potential multiple edges and loops).
Our first lemma says that in any multigraph either some multiedges 'cluster' together or many pairs of multiedges are far apart, or one can find a good k-matching. We shall need the following inequality.
The above inequality is easily seen to be equivalent to Let k be a natural number and ε 1 , ε 2 be positive reals such that ε 1 + ε 2 ≤ 2 −k . Then, for sufficiently large M = M (k), if G is a multigraph on M multiedges, then at least one of the following conditions is satisfied.
1. There is a multiedge in G which is incident with at least ε 1 M multiedges; 2. There are at least ε 2 M 2 pairs of multiedges which are at distance greater than 1; 3. G contains a good k-matching.
Proof. We shall use induction on k. The base case when k = 1 is trivial -Condition 3 is always satisfied. Assume then that k ≥ 2 and the lemma is true for k − 1.
Suppose every multiedge is incident with at most ε 1 M multiedges and at most ε 2 M 2 pairs of multiedges are at distance greater than 1. We shall show that G contains a good k-matching. By an averaging argument there is a multiedge e which is at distance at most 1 from at least (1 − ε 2 )M − 1 multiedges. Let E ′ be the set of those multiedges which are at distance exactly 1 from e. It follows from our assumptions that By assumption, at most ε 2 M 2 of the multiedges in G ′ are at distance greater than 1. Therefore, since M ≤ M ′ +1 1−2 −k , for large enough M (and hence large enough M ′ ) we have that at most pairs of multiedges in G ′ are at distance greater than 1. Note that for k ≥ 2 one has where the last inequality is precisely Fact 3.1. Therefore, by the induction hypothesis, G ′ contains a good (k − 1)-matching. But since e is at distance 1 from any multiedge in G ′ , we also have a good k-matching in G.
Since we shall be operating with planar graphs, we single out the following corollary.
Corollary 3.3. Let M be a sufficiently large integer and let ε 1 , ε 2 be positive reals such that ε 1 + ε 2 ≤ 1 32 . If G is a planar multigraph with M multiedges then either G has a multiedge which is incident with at least ε 1 M multiedges or there are at least ε 2 M 2 pairs of multiedges at distance greater than 1.
Proof. If G contained a good 5-matching then it would contain a K 5 minor.
One strategy in the proof of our main theorem is to consider a suitable bipartition of our path-pairable planar graph, and to exploit the fact that any bipartite planar graph on n vertices has at most 2n − 4 edges. To exploit this last property we shall need ways of finding pairings of the vertices such that their corresponding edge-disjoint paths contribute 'many' edges to the bipartition. This is formalized in the following lemma. Proof. We say that a path in G is weak if it begins and ends in A, uses no edges inside B, and uses at most 2 edges between A and B. Now, let C := ⌈4ε −1 ⌉ and note that since ε ≤ 1/2 We claim that U x is small for every x ∈ U ; namely, it is easy to see that |U x | ≤ D C + Let us define an auxiliary graph G U with vertex set U where we join two vertices x, y provided y / ∈ U x (equivalently, x / ∈ U y ). It is easy to see that G U has a perfect matching (or 'almost' perfect, if |U | is odd; this makes no difference for us). Indeed, the degree of every vertex in G U is at least |U | − ε 2 n ≥ |U |/2, and therefore G U has a Hamilton cycle. Fix a perfect matching M in G U according to this Hamilton cycle and fix a pairing P of the vertices of G where each edge of M forms a pair. Finally, since G is path-pairable, choose a collection of edge-disjoint paths R that realize this pairing. Observe that any path from R must use an even number of edges between A and B. There are two types of edges in e = xy ∈ M with respect to this realization: either the x − y path in R is weak but dist(x, y) > C, or this x − y path uses at least 4 edges between A and B. Let where E 0 denotes the edges satisfying the former condition, E 1 the latter, and E 2 denotes the remaining edges. We claim that most edges are in E 1 . Indeed, observe that if e = xy ∈ E 2 , then the x − y path must use edges from B. By planarity we have e(B) < 3|B|, and therefore |E 2 | < 3εn. Using planarity again we have that e(A) < 3|A|.
On the other hand, for each edge in E 0 its path in R uses more than C edges, at most 2 of which are in the cut {A, B}, and none of which belong to B. Accordingly, since these paths are edge-disjoint, we have that e(A) ≥ (C − 2)|E 0 | and so It follows that since every path in R pairing an edge in E 1 contributes at least 4 edges between A and B, and these paths must be edge-disjoint, we have This completes the proof of Lemma 3.4.
Our final lemma allows us to quantify more precisely the degree distribution in any bipartite planar graph.
It follows that A ≤1 ≤ 2|A| + 2|B| − e (A, B). Finally, we see that We are now in a position to prove our main theorem. First, let us give a rough sketch of the proof. Let G be a path-pairable planar graph. We first partition the vertex set of G into the set A of vertices of small degree and the set B of vertices of large degree. We can apply Lemma 3.4 to find that there are many edges in this cut. We shall then show that most vertices in A have degree 2 in this bipartite graph. If Y ⊂ A denotes the vertices of degree 2, then we define a planar multigraph with vertex set B where we join x, y ∈ B whenever there is a v ∈ Y joined to precisely x and y. Now, using Corollary 3.3, we are able to either find a vertex of linear degree in B, or we can find many pairs of multiedges in our multigraph that are far apart. This, however, allows us to find a pairing which contributes to more than 2n edges between A and B, a contradiction to planarity.
We restate Theorem 1.2 for convenience. Theorem 1.2. There exists c ≥ 10 −10 10 such that if G is a path-pairable planar graph on n vertices then ∆(G) ≥ cn.
Proof. Suppose G is a path-pairable planar graph and fix some large constant D so that D −1 ≤ 8.5 · 10 −6 . Partition the vertex set of G into sets A and B, where B = {v ∈ V (G) : Since e(G) < 3n it easily follows that |B| ≤ 6D −1 n := εn.
Suppose that ∆(G) < cn, where c is sufficiently small (depending only on D) given by Lemma 3.4. More precisely, we may take c = ε 4D 2⌈4/ε⌉+1 . Our aim is to obtain a contradiction to the planarity of G, and so there must exist a vertex of degree at least cn. Of course, this is trivial if cn ≤ 1, so we shall assume throughout that n ≥ 1/c. By Lemma 3.4 (with U = A) we have that there are at least 2|A| − 16εn ≥ 2n − 18εn edges between A and B.
Next, we shall show that there is a large subset of A which induces a graph with maximum degree at most 2. To see this, let Let X ⊂ A t be the set of vertices in A t with at least 3 neighbours in A t . Since every vertex in A t has more neighbours in B t than in A t , we have that every vertex in X has at least 3 neighbours in B t . Therefore, by Lemma 3.5, |X| ≤ 2|B t |, e(X, B t ) ≤ 6|B t |, and there are at least e(A t , B t ) − n − 3|B t | ≥ e(A, B) − n − 3|B t | vertices in A t with exactly two neighbours in B t . Let A * = A t \ X and B * = B t ∪ X. Now we have that every vertex in A * has at most 2 neighbours in A * and |B * | ≤ 3|B t | ≤ 57εn, so |A * | ≥ n − 57εn.
We have to make sure we still have many vertices in A * with exactly two neighbours in B * . Notice that if a vertex v ∈ A t had two neighbours in B t and was not adjacent to any vertex in X then v ∈ A * and v still has exactly two neighbours in B * . Therefore we only have to worry about the vertices in A t which are adjacent to some vertices in X.
Hence there are at most 189εn vertices in A * which do not have degree 2 in B * .
We say that an edge uv ∈ G is bad if one of the followings holds: Now, since every vertex in A * has at most two neighbours in A * , each vertex in A * with degree not equal to 2 in B * contributes to at most two bad edges of Type II. As there are at most 189εn vertices in A * which do not have degree 2 in B * , it follows that there are at most 378εn bad edges of Type II. Let Y ⊆ A * be the set of vertices with degree exactly 2 in B * . We now define an auxiliary multi graph G B * in the following way. The vertex set of G B * is B * and for any two vertices x, y ∈ B * , join x to y by an edge for every v ∈ Y that is joined precisely to x and y. Proof. This is clear since the bipartite graph G[Y, B * ] between Y and B * is planar, and contracting edges preserves planarity.
Apply Corollary 3.3 to the multigraph G B * with ε 1 = ε 2 = 1/100. Notice that if an edge in G B * has degree bigger than 1 100 |Y | then one of its endpoints has degree at least 1 200 |Y |. However, recall that we initially assumed ∆(G) < cn, and certainly c ≤ 1/400 by our choice of D. Accordingly, since |Y | ≥ n − 189εn ≥ n/2, we obtain a vertex of degree at least 2c|Y | ≥ cn, a contradiction.
So we may assume that there are at least 1 100 |Y | 2 pairs of edges in G B * which are at distance greater than 1. Note that if H is any graph on n vertices with edge density at least δ, then it is easy to greedily find a matching of size at least δ 10 n. It follows that we may select a collection of pairwise disjoint pairs P in Y , such that |P| ≥ 1 1000 |Y | ≥ 1 2000 n, and such that for every {u, v} ∈ P, their corresponding edges in G B * are at distance greater than 1.
We need the following two claims.
Claim 3.8. Let P be a path contained in A * which has at least two vertices and does not contain any bad edges. Then every vertex v ∈ P has the same neighbourhood (of size 2) in Proof. This is immediate from the definiton of a bad edge. Claim 3.9. Let u, v ∈ Y be two vertices whose corresponding edges in G B * are at distance greater than 1. Then any path in G joining u and v either contains some bad edges, or uses at least 6 edges between A * and B * .
Proof. Suppose P is a path joining u and v which does not use any bad edges. By defintition and using claim 3.8, all vertices of V (P ) ∩ A * are in Y , it can not have an edge inside B * and it must use 2 or 4 edges between A * and B * . We may assume P uses 4 edges as the other case follows from the same argument. Let P = P 1 e 1 e 2 P 2 e 3 e 4 P 3 , where {e 1 , e 2 , e 3 , e 4 } are edges between A * and B * and P 1 , P 2 , P 3 are paths inside Y . From claim 3.8 appplied to P 1 , P 2 and P 3 we deduce that the edge of u in G B * is at distance at most 1 to the edge The proof of Theorem 1.2 is nearly complete. Indeed, since G is path-pairable, there are edge-disjoint paths joining every pair of P, and hence the pairs in P contribute to at least 6(|P| − 1233εn) edges between A * and B * .
Let P be the union of the vertices in P and let U = A * \P . Suppose first that |U | < 57εn.

It follows that
2|P| > (n − 57εn) − 57εn, so |P| > n/2 − 57εn. Then the above pairing contributes at least 6(n/2 − 1290εn) = 3n − 7740εn edges between A * and B * . But this is at least 2n whenever ε ≤ 7740 −1 which is guaranteed by our choice of D, a contradiction. Therefore, we may assume that |U | ≥ 57εn. By Lemma 3.4 (since c is small enough) there is a pairing of the vertices in U which contributes to at least 2|U | − 16 · 57εn = 2|U | − 912εn edges between A * and B * .
So by our choice of D we get that 8424ε ≤ 1 1000 , and so there are at least 2n edges between A * and B * , a contradiction to the planarity of G. It follows that there must exist a vertex of degree at least cn.

Final Remarks and Open Problems
It is worth observing that we only used that our graph did not contain a K 3,3 minor rather than the full planarity condition. In particular, note that our proof relied heavily on the fact that number of edges in any bipartite planar graph on n vertices is less than 2n. This constraint on the number of edges also holds for bipartite graphs which contain no K 3,3 minor, which allows us to carry through the rest of the proof.
We conjecture the following: Conjecture 4.1. For any t there exists a constant c = c(t) such that every path-pairable graph on n vertices without a K t minor must contain a vertex of degree at least cn.
Finally, recall that we defined ∆ p min (n) to be the minimum of ∆(G) over all n-vertex path-pairable planar graphs G. We have shown that ∆ p min (n) grows linearly in n; however, as mentioned in the Introduction, the constants in the upper and lower bounds are quite far apart. We close with the following problem.
Problem 4.2. Determine ∆ p min (n) for sufficently large n. We do not know if our construction yielding the upper bound of 2n/3 is optimal, and a significant improvement on our lower bound would be very interesting.