Face-degree bounds for planar critical graphs

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs, based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist. For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs.

χ (G) = ∆(G), then G is a class 1 graph, otherwise it is a class 2 graph. A class 2 graph G is critical, if χ (H) < χ (G) for every proper subgraph H of G. Critical graphs with maximum vertex-degree ∆ are also called ∆-critical. It is easy to see that critical graphs are 2-connected. A graph is planar if it embeddable into the Euclidean plane. A plane graph (G, Σ) is a planar graph G together with an embedding Σ of G into the Euclidean plane. That is, (G, Σ) is a particular drawing of G in the Euclidean plane.
Zhou [13] proved for each k ∈ {3, 4, 5} that if G is a planar graph with ∆(G) = 6 and G does not contain a circuit of length k, then G is a class 1 graph. Vizing's conjecture is confirmed for some other classes of planar graphs which do not contain some specific (chordal) circuits [1,9,10].  Figure 1: Graph G has two embeddings Σ, Σ such that F ((G, Σ)) = F ((G, Σ )).
Let G be a 2-connected planar graph, Σ be an embedding of G in the Euclidean plane and F (G) be the set of faces of (G, Σ). The degree d (G,Σ) (f ) of f is the length of its facial circuit. If there is no harm of confusion we also write d G (f ) instead of Clearly F ((G, Σ)) ≥ 3. As Figure 1 shows, F ((G, Σ)) depends on the embedding Σ. The local average face-degree of a 2-connected planar graph G is F * (G) = max{F ((G, Σ)) : (G, Σ) is a plane graph}.
This parameter is independent from the embeddings of G, and F * (G) ≥ 3 for all planar graphs. Let k be a positive integer and b k = sup{F (G) : G is a k-critical planar graph}, and b * k = sup{F * (G) : G is a k-critical planar graph}. If b k = ∞ and b * k = ∞, then we say that b k is a class 1 bound with respect to the average face-degree for k-critical planar graphs, and that b * is a class 1 bound w. r. t. the local average face-degree for k-critical planar graphs, respectively. If k = 1 or k ≥ 7, then every planar graph with maximum vertex-degree k is a class 1 graph, and therefore {F (G) : G is a k-critical planar graph} = {F * (G) : G is a k-critical planar graph} = ∅. Hence, b k and b * k do not exist in these cases. Therefore, we focus on k ∈ {2, 3, 4, 5, 6} in this paper. The main results are the following two theorems.
The next section states some properties of critical and of planar graphs. These results are used for the proofs of Theorems 1.1 and 1.2 which are given in Section 3.

Preliminaries
Let G be a 2-connected graph. A vertex v is called a k-vertex, or a k + -vertex, or a k − - and N (N (u, v)) = N (N (u)) ∪ N (N (v)).
Let (G, Σ) be a plane graph. A face f is called k-face, or a k + -face, or a k − -face, if respectively. We will use the following well-known results on critical graphs.
Lemma 2.1. Let G be a critical graph and e ∈ E(G). If e = xy, then d G (x) ≥ 2, and Lemma 2.2 (Vizing's Adjacency Lemma [7]). Let G be a critical graph and e ∈ E(G).
If e = xy, then x is adjacent to at least (∆(G) − d G (y) + 1) ∆(G)-vertices other than y.  We will use the following results on lower bounds for the number of edges in critical graphs.
Lemma 2.8. Let t be a positive integer and > 0.
2. There is a 2-critical planar graph G with F (G) > t.
Proof. The odd circuits are the only 2-critical graphs. Hence, the second statement and the first statement for k = 2 are proved. Let X and Y be two circuits of length n ≥ 4, where the indices are added modulo n. Consider an embedding, where Y is inside X. Add edges x i y i to obtain a planar cubic graph G with F * (G) = 1 3 (n + 8). Add edges x i y i+1 to obtain a 4-regular planar graph H with F * (H) = 1 4 (n + 9). Subdividing one edge in G and one in H yields a critical planar graph G n with ∆(G n ) = 3, and a critical planar n+2 . Now, the statements for 3-critical and 4-critical graphs follow. Examples of these graphs are given in Figure 2. The following lemma is implied by Euler's formula directly.   1. If k = 3, then F (G) < 8.
Proof. Let k = 3 and suppose to the contrary that F (G) ≥ 8. With Lemma 2.9 and , a contradiction. The other statements follow analogously with the Lemma 2.9 and Theorem 2.6 (k ∈ {4, 5}) and Theorem 2.7 (k = 6).

Theorem 1.2
The statement for k ∈ {2, 3, 4} and for the lower bound for b * 5 follow from Lemma 2.8. It remains to prove the upper bounds for b * 5 and b * 6 . The result for b * 5 is implied by the following theorem. Proof. Suppose to the contrary that F * (G) = r > 7 + 1 2 . Let Σ be an embedding of G into the Euclidean plane and F * (G) = F ((G, Σ)). Let V = V (G), E = E(G), and F be the set of faces of (G, Σ). We are going to proceed a discharging procedure in G, by which we eventually deduce a contradiction. Define the initial charge ch in G as ch(x) = d G (x) − 4 for x ∈ V ∪ F . Euler's formula |V | − |E| + |F | = 2 can be rewritten as: We define suitable discharging rules to change the initial charge function ch to the which is the desired contradiction.
It remains to check the final charge for all x ∈ V ∪ F .
then v receives at least 1 3 − 4 3r −6 in total from its incident faces by Claim 3.2.1. By Lemmas 2.1 and 2.2, v has three 4 + -neighbors, and two of them have degree 5.
If v has a 4-neighbor, then by R2.
5r −12 . It remains to consider the case when v has five 4 + -neighbors. In this case it follows The result for k = 6 in Theorem 1.2 is implied by the following theorem.
Proof. Suppose to the contrary that F * (G) > 3 + 2 5 . Let Σ be an embedding of G into the Euclidean plane and F * (G) = F ((G, Σ)). We have It remains to show that ch * (x) ≥ 0 for every x ∈ V (G) ∪ F (G).
. First we consider the case when v is heavy. On one hand, since F ((G, Σ)) > 3 + 2 5 , it follows that either v is incident with a 5 + -face and another 4 + -face or v is incident with at least three 4-faces. In both cases, v receives at least 13 10 in total from its incident faces by R1. On the other hand, we claim that v sends at most 3 10 out in total. If v is adjacent to a bad-light vertex u, then all other neighbors of v have degree 6 by Lemma 2.3. Hence, v sends 3 10 to u by R2 and nothing else to its other neighbors. If v is adjacent to no bad-light vertex, then v has at most three good-light neighbors by Lemma 2.2. Hence, v sends 1 10 to each good-light neighbor by R2 and nothing else to its other neighbors. Therefore, ch * (v) ≥ ch(v) + 13 10 − 3 10 = 0. Second we consider the case when v is not heavy. In this case, v sends no charge out.
If v is incident with a 6 + -face, then v receives at least 1 from this 6 + -face by R1. This gives ch * (v) = ch(v) + 1 = 0. If v is incident with at least two 4 + -faces, then v receives at least 1 2 from each of them by R1. This gives ch * (v) = ch(v) + 1 2 + 1 2 = 0. We are done in both cases above. Hence, we can assume that v is incident with no 6 + -face and with at most one 4 + -faces, that is, v is light. From F ((G, Σ)) > 3 + 2 5 it follows that v is incident to a face f v such that d G (f v ) ∈ {4, 5}.
If d G (f v ) = 4, then v has degree 2. It follows that the two neighbors of v are heavy.
Thus, v receives 1 2 from f v by R1 and 3 10 from each neighbor by R2. Hence, ch * (v) = ch(v) + 1 2 + 3 10 + 3 10 > 0. If d G (f v ) = 5, then v receives 4 5 from f v . If v is not a bad-light 4-vertex, then by Lemma 2.2 every neighbor of v has degree 5 or 6. Hence, both of the two neighbors of v contained in f v are heavy. By R2, each of them sends charge at least 1 10 to v, and therefore, ch * (v) ≥ ch(v) + 4 5 + 1 10 + 1 10 = 0. If v is a bad-light 4-vertex, then at least one of the two neighbors of v contained in f v is heavy. Thus, this heavy neighbor sends charge 3 10 to v, and therefore, ch * (v) ≥ ch(v) + 4 5 + 3 10 > 0. Seymour's exact conjecture [6] says that every critical planar graph G is overfull,

Concluding remarks
i.e. |V (G)| is odd and |E(G)| = ∆(G) 1 2 |V (G)| + 1. If this conjecture is true for k ∈ {3, 4, 5}, then b k is equal to the lower bound given in Theorem 1.1. It is also not clear whether b k and b * k or F (G) and F * (G) are related to each other, respectively. By Proposition 3.1, F (G) has an upper bound for every critical planar graph G.
However, this is not always true for class 2 planar graphs. Similarly, Theorems 3.2 and 3.3 can not be generalized to class 2 planar graphs.