Asymptotic Equivalence of Hadwiger's Conjecture and its Odd Minor-Variant

Hadwiger's conjecture states that every $K_t$-minor free graph is $(t-1)$-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd $K_t$-minor is $(t-1)$-colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form $Ct$ for some constant $C>0$ exists. We show that if every graph without a $K_t$-minor is $f(t)$-colorable, then every graph without an odd $K_t$-minor is $2f(t)$-colorable. Using this, the recent $O(t\log\log t)$-upper bound of Delcourt and Postle for the chromatic number of $K_t$-minor free graphs directly carries over to the chromatic number of odd $K_t$-minor-free graphs. This (slightly) improves a previous bound of $O(t(\log \log t)^2)$ for this problem by Delcourt and Postle.


Introduction
Given a number t ∈ N, a K t -expansion is a graph consisting of vertex-disjoint trees (T s ) t s=1 and exactly one connecting edge between any pair of trees T s , T s ′ for distinct s, s ′ ∈ {1, . . . , t}. A graph is said to contain K t as a minor or to contain a K t -minor if it admits a subgraph which is a K t -expansion. Hadwiger's conjecture, which may well be seen as one of the most central open problems in graph theory, states the following relation between minor containment and the chromatic number of graphs. Conjecture 1 (Hadwiger 1943, [3]). If G is a graph which does not contain K t as a minor, then χ(G) ≤ t − 1.
Hadwiger's conjecture and many variations of it have been studied in past decades, a very good overview of the developments and partial results until about 2 years ago is given in the survey article [5] by Seymour. The best known asymptotic upper bound on the chromatic number of K t -minor free graphs for a long time remained of magnitude O(t √ log t), as proved independently by Kostochka [13] and Thomason [20] in 1984. However, recently there has been progress. First, in 2019, Norine, Postle and Song [14] broke the t √ log t barrier by proving an upper bound of the form O(t(log t) β ) for any β > 1 4 . Subsequently, there have been several significant improvements of this bound and related results [15], [17], [18]. The following state of the art-bound was proved recently by Delcourt and Postle in [1].

Theorem 1. The maximum chromatic number of K t -minor free graphs is bounded from above by a function in O(t log log t).
A strengthening of Hadwiger's conjecture to so-called odd minors was conjectured by Gerards and Seymour in [4]. A K t -expansion H certified by a corresponding collection of vertex-disjoint trees (T s ) t s=1 is said to be odd, if there exists an assignment of two colors {1, 2} to the vertices of H in such a way that every edge contained in one of the trees T s with s ∈ {1, . . . , t} is bichromatic (i.e., has different colors at its endpoints), while every edge joining two distinct trees is monochromatic (i.e., has the same color at its endpoints).
Finally, we say that a graph contains K t as an odd minor or that it contains an odd K t -minor if it contains a subgraph which is an odd K t -expansion. [4]). If G is a graph which does not contain K t as an odd minor, then χ(G) ≤ t − 1.

Conjecture 2 (Gerards and Seymour
Just as for Hadwiger's conjecture, the asymptotic growth of the best-possible upper bound on the chromatic number of graphs without an odd K t -minor has been studied. First, Geelen, Gerards, Reed, Seymour and Vetta proved in [2] that every graph with no odd K t -minor is O(t √ log t)-colorable. A shorter proof for the same result was given by Kawarabayashi in [8].
Subsequently, an asymptotical improvement of this upper bound to O(t(log log t) β ) for any β > 1 4 was achieved by Norine and Song in [16]. This was improved further to O(t(log log t) 6 ) by Postle in [19]. Very recently the exponent of the log log t factor was further improved by Delcourt and Postle in [1], resulting in an O(t(log log t) 2 )-bound. Many further results on odd K t -minor free graphs are known, we refer to [6,7,9,10,11,12] for some additional references. The purpose of this note is to show that asymptotically, the maximum chromatic number of K t -minor free graphs and the maximum chromatic number of odd K t -minor free graphs differ at most by a multiplicative factor of 2.
Theorem 2. Let t ∈ N and let f (t) be an integer such that every graph not containing K t as a minor is f (t)-colorable. Then every graph not containing K t as an odd minor is 2f (t)-colorable. Theorem 2 has a very simple proof, given in Section 2 below. It is useful in the sense that any progress made towards a better asymptotic upper bound on the chromatic number of K t -minor free graphs carries over, without further work and only at the prize of a constant multiplicative factor, to odd K t -minor free graphs. In particular, Theorem 1 together with Theorem 2 directly yields the following (slight) asymptotical improvement of the O(t(log log t) 2 )-upper bound on the chromatic number of odd K t -minor free graphs by Delcourt and Postle.

Corollary 3.
The maximum chromatic number of odd K t -minor free graphs is bounded from above by a function in O(t log log t).

Proof of Theorem 2
The proof is based on the following lemma.

Lemma 4.
Let G be a graph. Then there exists n ∈ N and a partition of V (G) into n non-empty sets X 1 , . . . , X n such that the following hold: • for every 1 ≤ i ≤ n, the graph G[X i ] is bipartite and connected, • for every 1 ≤ i < j ≤ n, either there are no edges in G between X i and X j , or there exist u 1 , u 2 ∈ X i and v ∈ X j such that u 1 v, u 2 v ∈ E(G) and u 1 and u 2 lie on different sides of the bipartition of G[X i ].
Proof. Let us define the partition X 1 , X 2 , . . . of V (G) inductively as follows: Suppose that for some integer i ≥ 1, all the sets X k with 1 ≤ k < i have been defined already, and do not yet form a partition, i.e., 1≤k<i X k = V (G). We now choose X i as an inclusion-wise maximal set among all subsets X ⊆ V (G) \ 1≤k<i X k which satisfy that G[X] is a bipartite and connected graph. Note that X i = ∅, since for every vertex x ∈ V (G) \ 1≤k<i X k , the graph G[{x}] is bipartite and connected.
Since we are adding a non-empty set to our collection of pairwise disjoint subsets of V (G) at each step, the above procedure eventually yields a partition X 1 , . . . , X n of V (G) for some n ∈ N. By definition, we have that G[X i ] is bipartite and connected for i = 1, . . . , n, and hence what remains to show is the second property of the partition stated in the lemma.
So let i, j be given such that 1 ≤ i < j ≤ n, and suppose that there exists at least one edge e ∈ E(G) between X i and X j . Denote e = uv with u ∈ X i and v ∈ X j . Let {A; B} be the unique bipartition of G[X i ]. We claim that v must have a neighbor u 1 ∈ A and a neighbor u 2 ∈ B, which then yields the statement claimed in the lemma. Indeed, suppose not, and suppose w.l.o.g. that v is not adjacent to any vertex in A (the case that v has no neighbor in B is of course symmetric). Then also the graph G[X i ∪ {v}] is bipartite and connected: It is connected since G[X i ] is connected and because of the edge uv, and it is bipartite since {A ∪ {v}; B} forms its unique bipartition. However, putting X := X i ∪ {v} ⊆ V (G) \ 1≤k<i X k , this contradicts the definition of X i as an inclusion-wise maximal subset of V (G) \ 1≤k<i X k inducing a bipartite and connected subgraph.
We can now easily deduce Theorem 2.
Proof of Theorem 2. Let t ∈ N and suppose that f (t) is an integer such that every K t -minor free graph is f (t)-colorable. Let G be any given graph without an odd K t -minor, and let us prove that χ(G) ≤ 2f (t).
We apply Lemma 4 to G and obtain a partition X 1 , . . . , X n of V (G) with properties as stated in the lemma. Let H be defined as the graph with vertex-set {1, . . . , n} and which has an edge between distinct vertices i and j if and only if there exists at least one edge in G between X i and X j . It follows from the statement of the lemma that for every edge ij ∈ E(H) with i < j, there exist u 1 , u 2 ∈ X i and v ∈ X j such that u 1 v, u 2 v ∈ E(G) and Next, we will show that χ(H) ≤ f (t) by proving that H does not contain K t as a minor. Suppose towards a contradiction that H contains a subgraph which is a K t -expansion, i.e., there exist vertex-disjoint trees (T s ) t s=1 contained in H and for every pair {s, s ′ } ⊆ {1, . . . , t} an edge e(s, s ′ ) ∈ E(H) with endpoints in T s and T s ′ .
For every fixed s ∈ {1, . . . , t}, let us consider the subgraph G s := G i∈V (Ts) X i of G. This is a connected graph because G[X i ] is connected for every i ∈ V (T s ), since T s is connected, and since by definition of H for every edge ij ∈ E(T s ) there exists at least one connecting edge between X i and X j in G. In particular, G s contains a spanning tree T G s which has the property that T G s [X i ] forms a spanning tree of G[X i ], for every i ∈ V (T s ). The trees (T G s ) t s=1 in G defined as above are pairwise vertex-disjoint. Let us denote by c : Subproof. By assumption, there exists e(s, s ′ ) ∈ E(H) which connects a vertex in i ∈ V (T s ) to a vertex j ∈ V (T s ′ ). Possibly after relabelling assume w.l.o.g. 1 ≤ i < j ≤ n. Then the second property of the partition X 1 , . . . , X n guaranteed by Lemma 4 yields the existence of vertices , and such that u 1 and u 2 lie on different sides of the unique bipartition of G[X i ]. Since u 1 , u 2 ∈ X i ⊆ V (T G s ) and since by our choice of T G s the graph T G s [X i ] forms a spanning tree of G[X i ], it follows that u 1 and u 2 also must be on different sides in the unique bipartition of T G s [X i ]. In particular, c(u 1 ) = c(u 2 ), which implies that c(u r ) = c(v) for some r ∈ {1, 2}. Now the edge f r ∈ E(G) connects the vertex u r in T G s with the vertex v in T G s ′ , and is monochromatic with respect to c. This proves the subclaim with f (s, s ′ ) := f r .
It follows directly from the previous claim that the union of the vertex-disjoint trees (T G s ) t s=1 , joined by the edges f (s, s ′ ) for every pair {s, s ′ } ⊆ {1, . . . , t}, forms an odd K t -expansion contained in G. This contradicts our initial assumption that G does not contain K t as an odd minor. Hence, our initial assumption was wrong, and we have established that H is K t -minor free. It now follows that χ(G) ≤ 2χ(H) ≤ 2f (t), as required. This concludes the proof of the theorem.