Maximum average degree of list-edge-critical graphs and Vizing’s conjecture

Vizing conjectured that χ ′ ℓ ( G ) ≤ ∆ + 1 for all graphs. For a graph G and nonnegative integer k , we say G is a k -list-edge-critical graph if χ ′ ℓ ( G ) > k , but χ ′ ℓ ( G − e ) ≤ k for all e ∈ E ( G ) . We use known results for list-edge-critical graphs to verify Vizing’s conjecture for G with mad ( G ) < ∆+32 and ∆ ≤ 9


Introduction
We consider only simple graphs in this paper. It will be convenient for us to define for a graph G, the vertex set V x = {v ∈ V (G) | d(v) = x} and the set V [x,y] An edge-coloring of G is a function which maps one color to every edge of G such that adjacent edges receive distinct colors. A k-edge-coloring of G is an edge-coloring of G which maps a total of k colors to E(G). The chromatic index χ ′ (G) is the minimum k such that G is k-edge-colorable. Vizing's Theorem [10] gives us χ ′ (G) ≤ ∆ + 1 for all graphs G where ∆ is the maximum degree of G.
We are interested in a variation of edge-coloring called list-edge-coloring. A list-edge-coloring is an edge-coloring with the extra constraint that each edge can only be colored from a preassigned list of colors. Specifically, we say an edge-list-assignment of G is a function which maps a set of colors to every edge in G. If L is an edge-list-assignment of G, then we refer to the set of colors mapped to e ∈ E(G) as the list, L(e). We say that G is L-colorable if G can be properly edgecolored with every edge e receiving a color from L(e). We say that G is k-list-edge-colorable if G is L-colorable for all L such that |L(e)| ≥ k for all e ∈ E(G). We note this concept is referred to as k-edge-choosable in other papers. The list-chromatic index, χ ′ ℓ (G), is the minimum k such that G is k-list-edge-colorable. So, we want to achieve a list-edge-coloring for all list-assignments L with minimal list-size k.
It is easy to see that χ ′ ℓ (G) ≥ χ ′ (G) ≥ ∆ for all graphs. The List-Edge Coloring Conjecture proposes that χ ′ ℓ (G) = χ ′ (G), but this has only been verified for a few special families of graphs, such as Galvin's result for the family of bipartite graphs [6]. In this paper, we will focus on a relaxation of the LECC proposed by Vizing.
The average degree of a graph G is ad( is the maximum of the set of average degrees of all subgraphs G. Motivated by Vizing and the List Edge Coloring Conjecture, Woodall conjectured [11] if G has mad(G) < ∆ − 1, then χ ′ ℓ (G) = ∆. Together with Borodin and Kostochka, Woodall [2] was able to verify his conjecture when mad(G) < √ 2∆. We say that a graph G is k-list-edge-critical if χ ′ ℓ (G) > k, and χ ′ ℓ (G−e) ≤ k for all e ∈ E(G). By taking advantage of known results for list-edge-critical graphs, we relax Woodall's conjecture by bounding ∆(G) ≤ 9 to verify Conjecture 1 when mad(G) < ∆(G)+3 2 .

Main Result
In 1990, Borodin verified Conjecture 1 for planar graphs with ∆ ≥ 9 (see [3]). This was improved to planar graphs with ∆ ≥ 8 by Bonamy in 2015 (see [1]). In 2010, before Bonamy's result, Cohen and Havet wrote a new proof of Borodin's theorem which reduced the argument to about a single page (see [4]). Their new proof used the minimality of list-edge-critical graphs and a clever discharging argument. We state one of their lemmas below.
Lemma 2.1, together with Borodin, Kostochka, Woodall's generalization [2] of Galvin's Theorem, were used to prove the following lemma. This lemma is listed as Lemma 9 in [7] and was used to achieve edge-precoloring results.
Lemma 2.2 (Harrelson, McDonald, Puleo [7]). Let a 0 , a, b 0 ∈ N such that a 0 > 2, b 0 > a, and We apply Lemma 2.2 directly to graphs of bounded maximum average degree to prove our main result.
Proof. Let m = ∆+3 2 and assign integers, which we will call an initial charge, to every vertex and an artificial, global pot P . We denote and define these initial charges as follows: . We will apply a discharging step and denote α ′ (v) as the final charge for v ∈ V (G) after discharing. We will also use α ′ (P ) and α ′ (G) to denote the final charges of P and G, respectively, after the discharging step. To get a contradiction, We note that this theorem is known for ∆ ≤ 4 so we may assume 5 ≤ ∆ ≤ 9. For each of these values of ∆, we provide Tables 1 through 5. Each table provide a list of triples (a 0 , a, b 0 ) and their resulting inequalities from Lemma 2.2. Each table also presents the discharging step and verifies α ′ (v) ≥ m for all v ∈ V (G). We let x i be the sum of coefficients of V i from the first table.
For all values of ∆, we discharge in the following way; If For all values of ∆, we verify α ′ (P ) > 0 by using only strict inequalities and noting the lesser side of every inequality only contains vertices with degree less than m and the greater side every inequality only contains vertices with degree greater than m. This means more charge is put into P than is taken from P due to how we defined x i in our discharging step.
If ∆ = 9, then we consider the ordered triples in the form of (a 0 , a, b 0 ) and the system of inequalities resulting from Lemma 2.2 as displayed in Table 1. We note that the final charge of P is positive since adding all inequalities together yields: The final charges from Table 1 gives This is a contradiction for ∆ = 9. We proceed through the remaining values of ∆ using the same argument. We present a table for each value of ∆. Each table displays inequalities resulting from Lemma 2.2 and each table displays the discharging step to verify α ′ (v) > m and α ′ (P ) > 0. Note that, for ∆ = 8, we multiply the first inequality by 1/2.

Conclusion
The application of Lemma 2.2 can be improved for some values of ∆(G) presented in Theorem 2.1 to yield slightly greater values of mad(G). We can also apply Lemma 2.2 to any value of ∆(G), but this will lower the bound on mad(G). Specifically, we can find optimum values of mad(G) given ∆(G) for graphs of higher max-degree by "reverse-engineering" the inequalities of Lemma 2.2 as shown in the following example for ∆(G) = 10.
Proof. Let mad(G) < m for some m, let α(P ) = 0, and let α(v) = d(v) for all v ∈ V (G). We wish to determine the largest number m such that α ′ (P ) > 0 and α ′ (v) ≥ m for all v ∈ V (G). We begin by presenting a table of triples and their resulting inequalities from Lemma 2.2; however, we multiply each inequality by an arbitrary constant.
As in Theorem 2.1, we let x i be the sum of coefficients of V i from this table. We will let "highdegree" vertices give charge to P while "low-degree" vertices take charge from P in the rules that follow. If deg(v) = i ≥ ⌈ 1 2 ∆ + 2⌉, then v gives x i to P . If deg(v) = i ≤ ⌊ 1 2 ∆ + 1⌋, then v takes x i from P . This yields the list of final charges displayed in Table 6. We set each final charge greater than or equal to m.
Increasing the constants c 1 ,c 2 ,c 3 ,c 4 increases the final charge of our "low-degree" vertices, but decreases the final charge of our "high-degree" vertices. We need all final charges to be greater than or equal to m so we must chose m carefully. While all vertices in V [7,10] give charge away, the vertices in V 10 give the most, meaning inequality E has the strictest bound on m. With this in mind, we can find an optimal bound for m by adding inequalities in the following way: 2E + A + B + C + D =⇒ 38 + 0x 1 + 0x 2 + 0x 3 ≥ 6m =⇒ 19 3 ≥ m We can now use this bound and the inequalities of the "low-degree" vertices from Table 6 to solve for c 1 ,c 2 ,c 3 ,c 4 .
We have shown that α ′ (v) ≥ 19 3 for our "low-degree" vertices in V [3,6] . We only need to verify the values of c 1 , c 2 , c 3 , c 4 , and m give us appropriate inequalities for the "high-degree" vertices.