Corrigendum to “Describing quasi-graphic matroids” [European J. Combin. 85 (2020) 103062] (European Journal of Combinatorics (2020) 85, (S0195669819301635), (10.1016/j.ejc.2019.103062))

Abstract

In our paper Describing quasi-graphic matroids [1], we defined the notion of a bracelet function for a biased graph, and the notion of a tripartition of the cycles of a biased graph as a refinement of the bipartition of its cycles as balanced or unbalanced. Our main theorem, Theorem 1.1, provided a characterisation of quasi-graphic matroids in terms of these descriptions. However, Theorem 1.1 does not necessarily hold for disconnected matroids, so as stated is false. In this note we state and prove the following corrected version of Theorem 1.1. Theorem 1 Let [Formula presented] be a connected matroid and let [Formula presented] be a biased graph with [Formula presented]. The following are equivalent. (1) There is a proper bracelet function [Formula presented] for [Formula presented] such that [Formula presented].(2) There is a proper tripartition [Formula presented] of the cycles of [Formula presented] such that [Formula presented].(3) [Formula presented] is quasi-graphic with framework [Formula presented] and [Formula presented] is the set of cycles of [Formula presented] that are circuits of [Formula presented]. Our new Theorem 1 differs from Theorem 1.1 of [1] only by the addition of the condition that [Formula presented] be connected. The class of quasi-graphic matroids is closed under direct sums, but in general neither the bracelet function construction nor the triparition construction permit a direct sum. Let [Formula presented] be the direct sum of a quasi-graphic matroid [Formula presented] that is not frame and a quasi-graphic matroid [Formula presented] that is not lifted-graphic. Let [Formula presented] be a framework for [Formula presented] and let [Formula presented] be a framework for [Formula presented], and let [Formula presented] be the disjoint union of [Formula presented] and [Formula presented]. Then [Formula presented] is clearly a framework for [Formula presented], but clearly there is no proper bracelet function [Formula presented] nor proper tripartition [Formula presented] of the cycles of [Formula presented] for which [Formula presented] or [Formula presented]. The statement that we in fact prove in [1] as Theorem 1.1 is the following. Theorem 2 Let [Formula presented] be a matroid and let [Formula presented] be a biased graph with [Formula presented]. Consider the following statements. (1) There is a proper bracelet function [Formula presented] for [Formula presented] such that [Formula presented].(2) There is a proper tripartition [Formula presented] of the cycles of [Formula presented] such that [Formula presented].(3) [Formula presented] is quasi-graphic with framework [Formula presented] and [Formula presented] is the set of cycles of [Formula presented] that are circuits of [Formula presented].Statements [Formula presented] and [Formula presented] are equivalent; either of [Formula presented] or [Formula presented] implies [Formula presented]. If [Formula presented] is connected, then [Formula presented] implies [Formula presented] and [Formula presented]. In particular, if [Formula presented] is connected, then [Formula presented], [Formula presented], and [Formula presented] are equivalent. The error in the proof of Theorem 1.1 in [1] occurs in the final sentence of the proof that (3) implies (1). Here we relied on Theorem 2.1 of [1] to claim that the bracelet function [Formula presented] is proper. However, Theorem 2.1 requires the graph [Formula presented] to be connected. Nevertheless, without any condition that either [Formula presented] or [Formula presented] be connected, the proofs given in [1] that (1) implies (2), and that (2) implies (3) are correct. The implication (2) [Formula presented] (1) is proved in Theorem 2.3 of [1], also with no connectivity condition. The implications (3) [Formula presented] (1) and (3) [Formula presented] (2) therefore hold provided [Formula presented] is connected, regardless of whether or not [Formula presented] is connected. Theorem 2 is useful, but we would also like descriptions of quasi-graphic matroids that do not place conditions on their frameworks. Theorem 3 addresses this, allowing us to conclude that, provided [Formula presented] is connected, [Formula presented] is indeed proper, regardless of whether [Formula presented] is connected or not. This allows us to prove Theorem 1. Theorem 3 Let [Formula presented] be a biased graph, and let [Formula presented] be a bracelet function for [Formula presented]. If [Formula presented] is the set of circuits of a connected matroid, then [Formula presented] is proper. The following lemma is the only new fact we need to prove Theorem 3. We provide here a full proof of Theorem 3, though it is identical to the proof of Theorem 2.1 in [1], save for the addition of its final two sentences, which cite Lemma 4 to deal with the case that [Formula presented] is not connected. Lemma 4 Let [Formula presented] be a biased graph, let [Formula presented] be a bracelet function for [Formula presented]. Suppose [Formula presented] is the set of circuits of a connected matroid [Formula presented], but that [Formula presented] has two components [Formula presented] and [Formula presented]. Let [Formula presented] in [Formula presented] and [Formula presented] in [Formula presented] be a pair of unbalanced cycles. Then [Formula presented], so [Formula presented] is a circuit of [Formula presented]. Proof Suppose to the contrary that [Formula presented]. Let [Formula presented] be an element in cycle [Formula presented], [Formula presented]. As [Formula presented] is connected, there is a circuit of [Formula presented] containing both [Formula presented] and [Formula presented]. As there is no path in [Formula presented] linking [Formula presented] and [Formula presented], this circuit is a bracelet [Formula presented], where, for each [Formula presented], [Formula presented] is an unbalanced cycle in [Formula presented] containing [Formula presented]. Claim Let [Formula presented], [Formula presented], [Formula presented] be unbalanced cycles with [Formula presented] disjoint from [Formula presented] and where [Formula presented] and [Formula presented] share at least one edge. If [Formula presented] then also [Formula presented]. Proof of Claim Let [Formula presented] be an edge of [Formula presented], and let [Formula presented] be a circuit of [Formula presented] contained in [Formula presented] containing [Formula presented]. Let [Formula presented] be a circuit of [Formula presented] with [Formula presented]. We claim that [Formula presented] is in fact contained in [Formula presented]. For suppose to the contrary that [Formula presented] contains an edge [Formula presented] of [Formula presented]. Let [Formula presented] be the path in [Formula presented] containing [Formula presented] whose endpoints are in [Formula presented] but none of whose internal vertices are in [Formula presented]. As [Formula presented] has no vertex of degree one, all edges of [Formula presented] are contained in [Formula presented] and so in [Formula presented]. Since [Formula presented] is dependent in [Formula presented] while [Formula presented] is not, this implies that [Formula presented] properly includes a circuit of [Formula presented], a contradiction. Thus [Formula presented] contains no edge of [Formula presented], so [Formula presented] is contained in [Formula presented].□ As [Formula presented], by the claim, also [Formula presented]. Applying the claim again, we conclude [Formula presented].□ Proof of Theorem 3 Suppose for a contradiction that [Formula presented] is the set of circuits of a matroid and [Formula presented] is not proper. Then there are bracelets [Formula presented] and [Formula presented] adjacent in the bracelet graph of [Formula presented] with [Formula presented] and [Formula presented]. Let [Formula presented] and [Formula presented]. The subgraph [Formula presented] is either a theta subgraph, tight handcuffs, or a bracelet. In the case that [Formula presented] is a theta, tight handcuffs, or dependent bracelet, [Formula presented] contains a circuit [Formula presented]. If [Formula presented] is an independent bracelet, and there is a path [Formula presented] linking [Formula presented] and [Formula presented] while avoiding [Formula presented], then [Formula presented]; put [Formula presented]. Let [Formula presented]. Since [Formula presented], by the circuit elimination axiom there is a circuit in [Formula presented] contained in [Formula presented]. Since [Formula presented] has cyclomatic number two and contains [Formula presented] this set is independent, a contradiction. So assume [Formula presented] is an independent bracelet, and if there is a path linking [Formula presented] and [Formula presented], then this path meets [Formula presented]. Suppose first that such a path exists. Let [Formula presented] be a minimal path linking [Formula presented] and [Formula presented]. Then [Formula presented] is a circuit. Let [Formula presented]. By the circuit elimination axiom there is a circuit contained in [Formula presented]. But [Formula presented] has cyclomatic number two and contains [Formula presented] but no path from [Formula presented] to [Formula presented], so again this set is independent, a contradiction. The final possibility is that [Formula presented] forms an independent bracelet and there is no path in [Formula presented] linking [Formula presented] and [Formula presented]. But Lemma 4 implies that [Formula presented] is a circuit, so this is impossible.□ Proof of Theorem 1 The proofs that statement (1) implies (2), and that statement (2) implies (3), are identical to those given in the proof of Theorem 1.1 in [1]. The proof that (3) implies (1) is identical to that given in the proof of Theorem 1.1 in [1], except for the last sentence of the proof. In the final sentence, replace the reference to Theorem 2.1 with a reference to Theorem 3, so this sentence reads, “Since [Formula presented] is the set of circuits of [Formula presented], [Formula presented], and by Theorem 3, [Formula presented] is proper”.□ There are only two other results in [1] that are affected by the omission of the requirement that [Formula presented] be connected in Theorem 1.1. These are Theorems 4.2 and 4.4, which, while as stated are false, are true when the condition that [Formula presented] be connected is added to their statements. Theorems 5 and 6 are the corrected versions of Theorems 4.2 and 4.4, respectively, of [1]. Theorem 5 Let [Formula presented] be a connected quasi-graphic matroid, and suppose [Formula presented] is neither lifted-graphic nor frame. Then [Formula presented] is not representable over any field. Theorem 6 Let [Formula presented] be a framework for a connected matroid [Formula presented]. If [Formula presented] has a loop, then [Formula presented] is either lifted-graphic or frame. The proofs of Theorems 5 and 6 are identical to those given of Theorems 4.2 and 4.4, respectively, in [1], except that their references to Theorem 1.1 are now to Theorem 1. Disconnected counter-examples to Theorems 4.2 and 4.4 are easy to construct. Let [Formula presented] be a frame matroid that is not lifted-graphic, representable over a field [Formula presented] with a framework [Formula presented] with a loop, and let [Formula presented] be a lifted-graphic matroid that is not frame, representable over [Formula presented] with a framework [Formula presented] that has a loop. Let [Formula presented] be the disjoint union of [Formula presented] and [Formula presented]. Let [Formula presented] be the direct sum of [Formula presented] and [Formula presented]. Then [Formula presented] is clearly a framework for [Formula presented], [Formula presented] is clearly neither lifted-graphic nor frame, and [Formula presented] is clearly representable over [Formula presented]. Theorem 4.4 was invoked as part of our proof that the class of quasi-graphic matroids is closed under minors. A minor adjustment to the sentence following the statement of Theorem 4.5 is needed, which should read, “By Theorem 6 if [Formula presented] is a loop and [Formula presented] then the component of [Formula presented] containing [Formula presented] is either lifted-graphic or frame, and hence so is the corresponding component of [Formula presented]”. Subsequent results in [1] relying on Theorem 1.1 whose hypotheses include the condition that the graph under consideration be connected, remain, along with their proofs, valid. References in their proofs to Theorem 1.1 should be replaced by references to Theorem 2 as stated in this note. Results relying on Theorem 1.1 whose hypotheses require a connected matroid all hold; their proofs are corrected by replacing references to Theorem 1.1 with references to Theorem 1 of this note. The remaining required corrections to [1] are as follows. The sentence after Corollary 4.7 should be amended to, “Thus every connected quasi-graphic matroid has a connected framework”. In the proof of Theorem 6.1, the third sentence should be, “By Theorem 1, [Formula presented] for some graph [Formula presented] with proper tripartition [Formula presented] of its cycles, so by Corollary 4.7 [Formula presented] has a connected framework [Formula presented]”. The proof of Theorem 6.5 cites Theorem 1.1 for the fact that (1) implies (2). So this proof remains valid; for clarity this reference should be replaced by a reference to Theorem 2. Theorem 6.6 includes the condition that the matroid [Formula presented] be connected, but this condition is not required and should be removed. The proof of Theorem 6.10 uses Theorem 6.6 to obtain a triparition representation [Formula presented] for the biased-graphic matroid [Formula presented], and then applies Theorem 1.1 to conclude that [Formula presented] is quasi-graphic. As [Formula presented] is 2-connected, the proof is valid, though for clarity this reference should be replaced by a reference to Theorem 2.