On graphs double-critical with respect to the colouring number

The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph is removed at any step. An edge e of a graph G is said to be double-col-critical if the colouring number of G-V(e) is at most the colouring number of G minus 2. A connected graph G is said to be double-col-critical if each edge of G is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer k greater than 4 and any positive number r, there is a k-col-critical graph with the ratio of double-col-critical edges between 1- r and 1.


Introduction
All graphs considered in this paper are assumed to be simple and finite. 1 The cycle on n vertices is denoted by C n . The complete graph K n on n vertices is referred to as an n-clique. Let G denote a graph. The number of vertices in a largest clique contained in G is denoted by ω(G). The vertexconnectivity of G is denoted by κ(G). The number of vertices and edges in G is denoted by n(G) and m(G), respectively.
Given a vertex v in G, N (v, G) denotes the set of vertices in G adjacent to v; deg(v, G) denotes the cardinality of N (v, G), and it is referred to as the degree of v (in G). A vertex of degree 1 is referred to as a leaf. Given a subset S of the vertices of G, the subgraph of G induced by the vertices of S is denoted by G[S], and we let N (S, G) denote the set ∪ s∈S N (s, G) \ S. The square of a graph G, denoted by G 2 , is the graph obtained from G by adding edges between any pair of vertices of G which are at distance 2 in G. Given two graphs H and G, the complete join of G and H, denoted by G + H, is the graph obtained from two disjoint copies of H and G by joining each vertex of the copy of G to each vertex of the copy of H. The chromatic number of G is denoted by χ(G), while the list-chromatic number of G is denoted by χ (G). Let ψ denote some graph parameter. An edge e of G is said to be double-ψ-critical if ψ(G − V (e)) ≤ ψ(G) − 2. A connected graph G is said to be double-ψ-critical if each edge of G is double-ψ-critical. For brevity, we may also refer to double-χ-critical edges and graphs as, simply, double-critical edges and graphs, respectively.
The introduction of the concept of double-ψ-critical graphs in [11] was inspired by a special case of the Erdős-Lovász Tihany Conjecture [2], namely the special case which states that the complete graphs are the only double-critical graphs. We refer to this special case of the Erdős-Lovász Tihany Conjecture as the Double-Critical Graph Conjecture. The Double-Critical Graph Conjecture is settled in the affirmative for the class of graphs with chromatic number at most 5, but remains unsettled for the class of graphs with chromatic number at least 6 [4,10,12,13].
In [11], it was proved that if G is a double-χ -critical graph with χ (G) ≤ 4, then G is complete. It is an open problem whether there is a non-complete double-χ -critical graph with list-chromatic number at least 5.
The double-κ-critical graphs, which in the literature are referred to as contraction-critical graphs (since the vertex-connectivity drops by one after contraction of any edge), are well-understood in the case where κ is 4. Some structural results have been obtained for contraction-critical graphs with vertex-connectivity 5. (See [6,Sec. 4] for references on contraction-critical graphs.) Bjarne Toft 2 posed the problem of characterising the double-col-critical graphs. Here col denotes the colouring number which is defined in the paragraph below.
In this paper, we characterise the double-col-critical graphs with colouring number at most 5.
In the remaining part of this section, we define the colouring number and present some fundamental properties of this graph parameter.
The colouring number of a graph. Suppose that we are given a non-empty graph G and an ordering v 1 , . . . , v n of the vertices of G. Now we may colour the vertices of G in the order v 1 , . . . , v n such that in the ith step the vertex v i is assigned the smallest possible positive integer which is not assigned to any neighbour of v i among v 1 , . . . , v i−1 . This produces a colouring of G using at most max i∈{1,...,n} colours. Taking the minimum over the set S n of all permutations of {1, . . . , n}, we find that the chromatic number of G is at most The number in (1) is called the colouring number of G, and it is denoted by col(G). The colouring number of the empty graph K 0 is defined to be zero. By (1), col(G) ≤ ∆(G) + 1 for any graph G. The colouring number was introduced by Erdős and Hajnal [3], but equivalent concepts were introduced independently by several other authors. It can be shown (see, for instance, [15]) that the colouring number of any non-empty graph G is equal to max{δ(H) | H is an induced subgraph of G} + 1 (2) and that the colouring number can be computed in polynomial time [9]. The non-empty graphs with colouring number at most k + 1 are also said to be k-degenerate [8]. Thus, a non-empty graph G is k-degenerate if and only if there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k in the remaining graph is removed at any step. We may think of a k-degenerate graph as a graph that can be 'degenerated' to the empty graph by removing vertices of degree at most k. The colouring number is monotone on subgraphs, that is, if F is a subgraph of a graph G then col(F ) ≤ col(G). For ease of reference, we state the following elementary facts concerning the colouring number of graphs.   A graph G is said to be k-col-critical, or, simply, col-critical, if col(G) = k and col(F ) < k for every proper subgraph F of G. Similarly, a graph G is said to be k-col-vertex-critical, or, simply, col-vertex-critical, if col(G) = k and col(F ) < k for every induced proper subgraph F of G. It is easy to see that every connected r-regular graph is (r + 1)-col-critical.     Observation 5. Given any graph G, there is a col-critical subgraph F of G with col(G) = col(F ) = δ(F ) + 1. In particular, if G is col-critical then col(G) = δ(G) + 1.
Proof. Let F denote a minimal induced subgraph of G with col(F ) = col(G). This implies col(F ) < col(F ) for any induced proper subgraph F of F , in particular, F is a col-vertex-critical graph. Suppose col(F ) > δ(F ) + 1. Then there is some proper induced subgraph F of F with δ(F ) + 1 = col(F ), and so col(F ) ≥ col(F ), a contradiction. Hence col(F ) = δ(F )+1. If G is col-vertex-critical, then F = G, and the desired result follows.
The two following results may be of interest in their own right.
Proposition 7 (Pedersen [11]). For any two non-empty disjoint graphs G 1 and G 2 , the colouring number of the complete join G 1 + G 2 is at most and at least min{col(G 1 ) + n(J 2 ), col(G 2 ) + n(J 1 )} where, for each i ∈ {1, 2}, J i is any subgraph of G i with minimum degree equal to col(G i ) − 1.
If, in addition, col(G i ) = δ(G i ) + 1 for each i ∈ {1, 2} (in particular, if both G 1 and G 2 are col-vertex-critical), then the colouring number of the complete join G 1 +G 2 is equal to the minimum in (3).
A graph G is said to be decomposable if there is a partition of V (G) into two (non-empty) sets V 1 and V 2 such that, in G, every vertex of V 1 is adjacent to every vertex of V 2 . Given any graph G, we let V δ (G) denote the set of vertices of G of minimum degree in G. Clearly, V δ (G) is non-empty for any non-empty graph.
Proposition 8 (Pedersen [11]). Let G denote a decomposable graph. Then G is col-critical if and only if the vertex set of G can be partitioned into two sets V 1 and V 2 such that is an independent set of G 2 , and δ(G 1 ) + n(G 2 ) = δ(G 2 ) + n(G 1 ) or (ii) G 2 is an edgeless graph, and where Q denotes a smallest component of G 1 (in terms of the number of vertices).

Double-col-critical graphs
The analogue of the Double-Critical Graph Conjecture with χ replaced by col does not hold. For instance, the non-complete graph C 2 6 is 4-regular, 5-col-critical, and double-col-critical. Since C 2 6 is planar, it also follows that it is not even true that every double-col-critical graph with colouring number 5 contains a K 5 minor. (In [5], it was proved that every double-critical graph G with χ(G) ≤ 7 at least contains a K χ(G) minor.) It is easy to see that the square of any cycle of length at least 5 is a double-col-critical graph with colouring number 5.
Proof. Let G denote a double-col-critical graph. If there are no vertices in G, then we are done. Let v denote an arbitrary but fixed vertex of G. If there is no vertex in G adjacent to v, then we are done, since then, by the connectedness of G, G is just the singleton K 1 . Let u denote a neighbour of v. By Observation 4, we need to show col( contradictions Observation 3. This shows col(G−v) is strictly less than col(G), as desired.
Proof. Let G denote a double-col-critical graph, and define k := col(G). Then, by Observation 9, G is also col-vertex-critical, and so, by Observation 6, δ(G) = k − 1. If k ≤ 3, then the desired result follows immediately from Observation 2. Suppose k = 4. Then, for any edge e ∈ E(G), col(G − V (e)) ≤ 2 and so, by Observation 1, G − V (e) is a forest. Fix an edge xy ∈ E(G). If G − x − y contains no edges, then G is 2-degenerate and so col(G) ≤ 3, a contradiction. Let T denote a component of G − x − y with at least one edge, and let u and v denote two leafs of T .
Since, as noted above, δ(G) = 3, it follows that both u and v are adjacent to both x and y. If u and v are adjacent in T , then G[{u, v, x, y}] K 4 , and so, since G is also col-vertex-critical, G K 4 . Hence we may assume that u has a neighbour in This completes the proof.  It is easy to verify that the graphs Q 1 , Q 2 , and Q 3 in Figure 1 are double-col-critical and have colouring number 5. None of the graphs Q 1 , Q 2 , and Q 3 are squares of a cycle. We shall see that Q 1 , Q 2 , Q 3 , and the squares of the cycles of length at least 5 are all the double-col-critical graphs with colouring number 5. First a few preliminary observations. Observation 11. If G is a double-col-critical graph, then δ(G) = col(G) − 1 and every pair of adjacent vertices of G has a common neighbour of degree δ(G) in G.
Proof. Let G denote a double-col-critical graph. Then, by Observation 9, G is also col-vertexcritical, and so, by Observation 6, δ(G) = col(G) − 1. Let xy denote an arbitrary edge of G. Now, by the definition of the colouring number, G − x − y has minimum degree at most (col(G) − 2) − 1 which is equal to δ(G) − 2. This means that some vertex of V (G) \ {x, y}, say z, which has degree at least δ(G) in G has degree at most δ(G) − 2 in G − x − y. The only way this can happen is if z has degree δ(G) in G and is adjacent to both x and y in G. This completes the argument.
Observation 12. If G is a non-complete double-col-critical graph, then G does not contain a clique of order col(G) − 1.
Proof. Let G denote a non-complete double-col-critical graph. By Observation 9, G is col-vertexcritical, and so, since G is also non-complete, G cannot contain a clique of order more than col(G)−1. Also, by Observation 9, in particular, G contains a clique of order col(G), a contradiction.
Proposition 13. Every double-col-critical graph with colouring number at least 3 is 2-connected.
a contradiction. This shows that G must be 2-connected.
Double-col-critical graphs with colouring number 5.

Observation 14.
If G is a double-col-critical graph with colouring number 5 and ab ∈ E(G), then a or b has degree 4 in G.
We shall say that a k-neighbour of a vertex x is a neighbour of x of degree k.
Proof of Observation 14. Let G denote a double-col-critical graph with colouring number 5. Then col(G) = δ(G) + 1 = 5. Let ab denote an edge of G. Suppose that both a and b have degree greater than 4 in G. By Observation 11, there is a common 4-neighbour c of a and b. We shall make repeated use of Observation 11 and Observation 12. The latter observation implies that G contains no 4-clique. There is a common 4-neighbour d of a and c. Since ω(G) ≤ 3, d is not adjacent to b. This implies that there is a common 4-neighbour e of b and c and e is not identical to d. The vertex e is not adjacent to a. The vertex a has degree at least 5, and so the common 4-neighbour of c and d must be e. We note that {a, b, c, d, e} induce a subgraph of G of minimum degree 3. Hence Recently, the first author obtained a characterisation of what he called minimal critical graphs with minimum degree 4. It turns out that our double-col-critical graphs with colouring number 5 are such graphs, and so -using the characterisation of minimal critical graphs of minimum degree 4 -we obtain a characterisation of the double-col-critical graphs with colouring number 5.
In the following result, which is the main result of this paper, we let Q 1 , Q 2 , and Q 3 denote the graphs depicted in Figure 1. The graph Q 2 is the dual of the Herschel graph which is the smallest nonhamiltonian polyhedral graph.
In order to prove Theorem 15, we first need to introduce a bit of notation and state the abovementioned characterisation of minimal critical graphs with minimum degree 4.
For the remaining part of this section we shall be using the following notation. We shall let C denote the set of simple connected graphs of minimum degree at least 4. An edge e of a graph G in C is essential if the graph G − e obtained from G by deleting e is not in C, and let us call e critical if the graph G/e obtained by contracting e and simplifying is not in C. An edge e is essential if and only if e is a bridge or at least one of its endvertices has degree 4; and e is critical if and only if the endvertices of e have a common 4-neighbour or N (V (e), G) consists of three common neighbours of the endvertices of e. We are now interested in the minimal critical graphs in C, that is, graphs G ∈ C with the property that each edge of G is both essential and critical.
For the description of the minimal critical graphs in C, we shall consider a number of bricks, that is, any graph isomorphic to one of the following nine graphs: K 5 , K 2,2,2 , K − 5 , K − 2,2,2 , K 5 , K 2,2,2 , K 5 , K 2,2,2 , or K 3 which are depicted in Figure 2. Each brick comes together with its vertices of attachment: For K 5 and K 2,2,2 , this is an arbitrary single vertex, for the other seven bricks these are its vertices of degree less than 4. The remaining vertices of the brick are its internal vertices, and the edges connecting two inner vertices are called its internal edges. Observe that every brick B has one, two, or three vertices of attachment, and that they are pairwise nonadjacent unless B is the triangle, that is, K 3 .
It turns out that the minimal critical graphs from C are either squares of cycles of length at least 5, or they are the edge disjoint union of bricks, following certain rules. This is made precise in the following theorem.
Theorem 16 (Kriesell [7]). A graph is a minimal critical graph in C if and only if it is the square of a cycle of length at least 5 or arises from a connected multihypergraph H of minimum degree at least 2 with at least one edge and |V (e)| ∈ {1, 2, 3} for all hyperedges e by replacing each hyperedge e by a brick B e (see Figure 2) such that the vertices of attachment of B e are those in V (e) and at the same time the only objects of B e contained in more than one brick, and (TB) the brick B e is triangular only if each vertex x ∈ V (e) is incident with precisely one hyperedge f x different from e and the corresponding brick B fx is neither Figure 2: The nine bricks. Vertices of attachment are displayed solid.
and, for any other vertex y ∈ V (e) \ {x} and hyperedge f y containing y but distinct from e, we have Proof of Theorem 15. In order to prove the desired result, we prove the following equivalent statement.
A graph is double-col-critical with colouring number 5 if and only if it is the square of a cycle of length at least 5 or one of the three graphs obtained by taking the union of two graphs G 1 and G 2 such that G i K 5 or G i K 2,2,2 for i ∈ {1, 2} and x ∈ V (G 1 ) ∩ V (G 2 ) if and only if x has degree 2 in G 1 and degree 2 in G 2 .
The 'if'-part of the statement above is straightforward to verify and it is left to the reader.
Let G denote an arbitrary double-col-critical graph with colouring number 5. It follows from the definition of double-col-critical graphs, Observation 6, Observation 9, Observation 11, and Observation 14 that G has the following properties. By (a), G is in C. By (b), every edge of G is essential, and, by (c), every edge of G is critical. Thus, G is minimal critical in C, and so Theorem 16 applies. Suppose that G is not the square of a cycle. Then G has a representation by a multihypergraph H as described in Theorem 16. If e is a 1-hyperedge then the unique attachment vertex of the corresponding brick B e in G is a cutvertex of G, a contradiction to Proposition 13. Suppose that there exists a 2-hyperedge e with V (e) = {u, v}. If B e is K 5 or K 2,2,2 then B e − V (e) is an induced subgraph of G of minimum degree 3, and since the vertex u ∈ V (e) has only two neighbours in that subgraph, there is an edge in G − (V (B e ) \ V (e)), contradicting (d). If, otherwise, B e is K − 5 or K − 2,2,2 then, by Theorem 16, B e is an induced subgraph of G of minimum degree 3. By (a), there is a vertex in V (G) \ V (B e ), and it has a neighbour in V (G) \ V (B e ). This contradicts (d). Hence there are only 3-hyperedges in H.
Suppose that H contains a 3-hyperedge e for which the corresponding brick B e is triangular. It follows from (a) and (d) that some vertex q ∈ V (G) \ V (B e ) is adjacent to at least two vertices in V (B e ). The vertex q is not adjacent to all three vertices of V (B e ), since otherwise G[V (B e ) ∪ {q}] would induce a 4-clique in G which contradicts Observation 12. Let x and y denote the neighbours of q in V (B e ). By Theorem 16 (TB), x is incident to exactly one hyperedge f x different from e. Similarly, y is incident to exactly one hyperedge f y different from e. If f x = f y , then, by Theorem 16 (TB.ii), B fx is K 5 or K 2,2,2 , in particular, f x is a 2-hyperedge, a contradiction. Hence f x = f y and so, since q ∈ V (f x ) ∩ V (f y ), it follows from Theorem 16 (TB.i) that not both B fx and B fy are triangular bricks. The fact that f x and f y are distinct and q ∈ V (f x ) ∩ V (f y ) implies that q is an attachment vertex of both B fx and B fy . Since q is adjacent to x and both q and x are attachment vertices, it follows that B fx must be triangular. Similarly, B fy must be triangular, and so we have obtained a contradiction. This shows that each hyperedge in H is of the type K 5 or K 2,2,2 .
Let e denote an arbitrary 3-hyperedge of H. If there are two vertices x, y ∈ V (e) of degree exceeding 4 in G then G − (V (B e ) \ {x, y}) has minimum degree at least 3, contradicting (d) applied to any internal edge of B e . Therefore, if there is a vertex x ∈ V (e) of degree exceeding 4 in G, then the two vertices y, z ∈ V (e) \ {x} are incident with precisely one further 3-hyperedge f y and f z , respectively, both distinct from e. If f y = f z then one may argue as above that G−V (B e −x) has minimum degree at least 3, contradicting (d) applied to any internal edge of B e . Hence f y = f z =: f . Let w be the vertex in V (f ) \ {y, z}. If w = x then {w, x} forms a 2-separator, and otherwise w = x is a cutvertex as x has degree exceeding 4. In either case, G − (V (B e − x) ∪ V (B f )) has minimum degree at least 3, again contradicting (d).
Hence all vertices of attachment have degree 4 in G. Let e denote a 3-hyperedge in H, and let x, y, and z denote the vertices of V (e). Again let f x , f y , f z denote the unique 3-hyperedge distinct from e incident with x, y, z, respectively. If they are pairwise distinct then G − V (B e ) has minimum degree at least 3, contradiction to (d). If f := f y = f z = f x then let w be the vertex in V (f ) distinct from y, z. As f x = f , we have w = x, so that G − (V (e) ∪ V (f )) has minimum degree at least 3, contradicting (d), unless there is a vertex in V (f x ) adjacent to both x and w; in this latter case, the unique 3-hyperedge distinct from f incident with w must be f x , and so the vertex u in V (f x ) \ {w, x} is a cutvertex of G, a contradiction to Proposition 13. Hence f x = f y = f z , and the desired statement follows.
Given our success in characterising the double-col-critical graphs with colouring number 5, we venture to ask for a characterisation of the double-col-critical graphs with colouring number 6. If G is a double-col-critical graph, then G+K k is a double-col-critical graph with col(G+K k ) = col(G)+k (see Proposition 17). This implies that the graphs Q 1 +K 1 , Q 2 +K 1 , Q 3 +K 1 , and C +K 1 , where C is the square of any cycle of length at least 5, are all double-col-critical graphs with colouring number 6. These are not the only double-col-critical graphs with colouring number 6; the icosahedral graph is yet another double-col-critical graph with colouring number 6. This latter fact was also observed by Stiebitz [14, p. 323], although in a somewhat different setting. The standard 6-regular toroidal graphs obtained from the toroidal grids by adding all diagonals in the same direction have colouring number 7 and are double-col-critical.
Complete joins of double-col-critical graphs. In [5], it was observed that if G is the complete join G 1 + G 2 , then G is double-critical if and only if both G 1 and G 2 are double-critical. Next we prove that the 'if'-part of the analogous statement for double-col-critical graphs is true. The 'only if'-part is not true, as follows from considering the double-col-critical graph C 2 6 : We have C 2 6 C 4 + K 2 but neither C 4 nor K 2 is double-col-critical.
Proposition 17 and the fact that both C 2 6 and K t are double-col-critical immediately implies the following result, which, in particular, shows that, for each integer k ≥ 6, there is a non-regular double-col-critical graph with colouring number k.

Double-col-critical edges
In [5], Kawarabayashi, the second author, and Toft initiated the study of the number of doublecritical edges in graphs. In this section, we study the number of double-col-critical edges in graphs. Kawarabayashi, the second author, and Toft proved the following theorem for which we shall prove an analogue for the colouring number.
The complete join C n + K 1 of a cycle C n and a single vertex is referred to as a wheel, and it is denoted W n . If n is odd, we refer to W n as an odd wheel.
Theorem 19 (Kawarabayashi,Pedersen & Toft [5]). If G denotes a 4-critical non-complete graph, then G contains at most m(G)/2 double-critical edges. Moreover, G contains precisely m(G)/2 double-critical edges if and only if G is an odd wheel of order at least 6.
The following result is just a slight reformulation of Theorem 19.
Corollary 20. If G denotes a 4-chromatic graph with no 4-clique, then G contains at most m(G)/2 double-critical edges. Moreover, G contains precisely m(G)/2 double-critical edges if and only if G is an odd wheel of order at least 6.
Proof. Let G denote a 4-chromatic graph with no 4-clique. If e = xy is a double-critical edge in G, then e is a critical edge of G and x is a critical vertex of G. We remove non-critical elements from G until we are left with a 4-critical subgraph G . At no point did we remove an endvertex of a double-critical edge. Thus, the number of double-critical edges in G is equal to the number of double-critical edges in G . Clearly, G is a non-complete graph, and so, by Theorem 19, the number of double-critical edges in G is at most m(G )/2 which is at most m(G)/2. The second part of the corollary now follows easily.
The following result -which is an analogue of Theorem 19 with the chromatic number replaced by the colouring number -extends Observation 10.
Proposition 21. If G denotes a 4-col-critical non-complete graph, then G contains at most m(G)/2 double-col-critical edges. Moreover, G contains precisely m(G)/2 double-col-critical edges if and only if G is a wheel of order at least 6.
Lemma 22. If e and f are two double-col-critical edges in a 4-col-critical non-complete graph, then e and f are incident.
Suppose e is an arbitrary double-col-critical edge in G. Then col(G − V (e)) = 2 which, by Observation 1 (iii), means that G−V (e) is a forest containing at least one edge and, since δ(G) = 3, δ(G − V (e)) ≥ 1 and each leaf in G − V (e) is adjacent to both endvertices of e in G. Let u and v denote two leafs of G − V (e). Now, if G contains some double-col-critical edge f which is not incident to e, then G − V (f ) contains no cycles, since col(G − V (f )) = 2, and so f is incident to both u and v, which implies G[{u, v} ∪ V (e)] K 4 , a contradiction. This means that any two double-col-critical edges of G are incident, and the proof is complete.
Proof of Proposition 21. Let G denote a 4-col-critical non-complete graph. Then n(G) ≥ 5 and, by Observation 5, δ(G) = 3 which implies m(G) ≥ n(G) · δ(G)/2 ≥ 8. By Lemma 22, we only have to consider two cases: (i) G contains three incident double-col-critical edges xy, yz, and xz, or (ii) there is a vertex v ∈ V (G) such that every double-col-critical edge of G is incident to v. If (i) holds, then, since m(G) ≥ 8, the desired statement follows. Suppose (ii) holds. Then the number of double-col-critical edges in G is at most deg(v, G). We may assume that there is at least one double-col-critical edge, say, vw in G. Suppose G − v is disconnected. Then, since δ(G) = 3, each component of G − v has minimum degree at least 2, and so, in particular, some component of G − v − w has minimum degree at least 2. This, however, contradicts the fact that G − v − w is a forest. Hence G − v is connected. By Observation 3, col(G − v) ≥ 3 and so, by Observation 1 (iii), G − v contains a cycle. Hence G − v is a connected graph with at least one cycle, and so m(G − v) ≥ n(G − v). Thus, which implies that the number of double-col-critical edges is at most m(G)/2 and that the number of double-col-critical edges is equal to m(G)/2 only if deg(v, G) = n(G) − 1 and G − v is a cycle.
Conversely, if G is a wheel on at least five vertices, then it is easy to see that exactly m(G)/2 edges of G are double-col-critical. This completes the proof.
Let k denote some integer greater than 3. Let D k denote the 2k-cycle with vertices labelled Observation 23. For every integer k greater than 3, the graph F k , as defined as above, is a 5-colcritical graph with colouring number 5 in which all edges except v 1 v k−1 and u 1 u k−1 are double-colcritical. Proposition 24. For each integer p greater than 4 and positive real number , there is a p-colcritical graph G with the ratio of double-col-critical edges between 1 − and 1.
Proof. If p = 5, the desired result follows directly from Observation 23 by letting k tend to infinity. Let p denote an integer greater than 5 and a positive real number. Let k denote an integer a lot greater than p, and let G denote the graph obtained by taking the complete join of F k and K p−5 . Then, by Proposition 8 (ii), G is p-col-critical, and, since k is a lot greater than p, all but the edges v 1 v k−1 and u 1 u k−1 are double-col-critical in G. By letting k tend to infinity the ratio of double-col-critical edges in G will from a certain point onwards be between 1 − and 1. This completes the argument.
Proposition 24 means that there is no result corresponding to Proposition 21 for colouring numbers greater than 4.

Concluding remarks
By a theorem of Mozhan [10] and, independently, Stiebitz [12], K 5 is the only double-critical graph with chromatic number 5, that is, K 5 is the only 5-chromatic graph with 100% double-critical edges, but we do not know whether there are non-complete 5-chromatic graphs with the percentage of double-critical edges arbitrarily close to a 100. In [5], Kawarabayashi, the second author, and Toft conjectured that if G is a 5-critical non-complete graph, then G contains at most (2 + 1 3n(G)−5 ) m(G) 3 double-critical edges.
As we have seen, the story is a bit different for the colouring number. By Proposition 24 for p = 5, there are non-complete graphs with colouring number 5 with the percentage of double-colcritical edges arbitrarily close to a 100. This only makes Theorem 15 all the more interesting. By Theorem 15, we are able to distinguish between graphs with colouring number 5 having 99.99% double-col-critical edges and those that have a 100% double-col-critical edges.
The problem of obtaining a concise structural description of the double-col-critical graphs with colouring number k ≥ 6 remains open. Given any graph G, it can be decided in polynomial time whether or not G is double-col-critical, since the colouring number itself can be computed in polynomial time. Nevertheless, given the structural complexity of the double-col-critical graphs with colouring number 6 mentioned on page 10, it seems likely that even the problem of obtaining a concise structural description of the double-col-critical graphs with colouring number 6 is nontrivial.