Parsimonious Edge-coloring on Surfaces

We correct a small error in a 1996 paper of Albertson and Haas, and extend their lower bound for the fraction of properly colorable edges of planar subcubic graphs that are simple, connected, bridgeless, and edge-maximal to other surface embeddings of subcubic graphs.


Introduction and Background
Consider a graph G with maximum degree ∆(G), and assign colors to the edges. It is wellknown from Vizing's Theorem [7] that every simple graph with maximum degree ∆(G) = k may be properly edge-colored using either k or k + 1 colors. In general, most of the edges of a graph can be properly colored with ∆(G) colors, with few (if any) edges needing an additional color. The question then arises: What fraction of the edges of G is guaranteed to be properly ∆(G)-edge colorable?
We will focus on graphs for which ∆(G) = 3, namely the cubic (3-regular) and subcubic (highest degree 3) graphs. Let |E(G)| be the number of edges in G, let c(G) be the number of edges in a three-edge colorable subgraph of G with the largest possible number of edges, and let (c) if G is simple, subcubic, planar, bridgeless, connected, and edge-maximal (no edges may be added to G without violating one of the prior-listed conditions), then y(G) ≥ 6 7 − 2 35|E(G)| (a value in the range .8-.8571).
Note that this statement of (c) is a corrected version of Albertson and Haas's original statement; see Section 2.
While the bound in (a) is best possible for general simple cubic G (see the Petersen graph), restricting the class of graphs under consideration produces better bounds. For example, Rizzi [5] shows that if a subcubic graph G is triangle-free (but might have multiple edges), then y(G) ≥ 13 15 . Mkrtchyan and Steffen [4] show that if the girth of G is g, then y(G) ≥ g g+1 when g is even and y(G) ≥ g−1 g when g is odd. (Note that in the subcubic case, the girth of G must be at least 8 in order for this result to give a stricter bound than the 13 15 fraction for triangle-free graphs in [5].) Kamiński and Kowalik [3] show that a simple subcubic graph G has y(G) ≥ 13 15 unless it is a 5cycle with two adjacent chords (a result also shown in [2]), and that a subcubic multigraph G has y(G) ≥ 7 9 ≈ .7778 unless it is K 3 with an arbitrary edge doubled. (This is not much better than the bound of y(G) ≥ 3 4 given for an arbitrary subcubic multigraph in [1].) Restricting to snarks gives better bounds; Steffen [6] showed that the Petersen graph is the only snark with y(G) = 13 15 , and that snarks with at least 16 vertices have y(G) ≥ 11 12 ≈ .9167. In contrast to the structural approaches given above, Albertson and Haas extended (b) to (c) by restricting to planar graphs, or in other words, by using information about the possible embeddings of G. In Section 3, we generalize using embedding information about G; that is, we take a topological viewpoint. This is of interest because there are often constraints imposed on the colorability of a graph by the genus of that graph.

Correction to Albertson and Haas
Albertson and Haas' approach to (c) was as follows: First, vertices of degree one may be eliminated as their incident edges do not affect y(G). Then, examine facial boundaries. Any series of adjacent degree-two vertices may be replaced with a single one without improving the colorability of G. Thus, no face has two adjacent vertices of degree two.
Albertson and Haas additionally assumed that each face has at most one vertex of degree two, reasoning that joining a pair of non-adjacent degree-two vertices by an additional edge increases |E(G)| while leaving c(G) constant and thus decreases y(G). However, an added edge may be colorable so that c(G) and y(G) both increase. (Unfortunately, it is not known whether an example demonstrating this exists: Albertson and Haas conjectured ([1, p. 5]) that every planar bridgeless graph G with ∆(G) = 3 and at least two vertices of degree two has y(G) = 1. This problem is still open.) Thus an additional condition must be introduced for their Theorem 3 [1]: the embedding of G must be edge-maximal.

Non-planar Embeddings
Result (c) of [1] does not directly generalize to surfaces other than the sphere. Assuming bridgelessness for planar graphs eliminates monofacial edges. This does not hold for higher-genus surfaces; consider a cellular embedding of K 4 on the torus. See Figure 1 at left. It is bridgeless but Figure 1. At left, a cellular embedding of K 4 on the torus; at right, the same embedding with a monofacial edge subdivided.
has two monofacial edges. Unless we constrain ourselves to closed 2-cell embeddings, we may have an edge-maximal embedding with a degree-two vertex v incident to two monofacial edges; in this case, v is not shared between two faces, but occurs twice on the same face. (For an example, consider the toroidal embedding of K 4 and subdivide one of the monofacial edges; this is shown at right in Proof. Let V 2 and V 3 be the number of degree-two and degree-three vertices of G, respectively. Additionally, let E = |E(G)| and F be the number of faces in the embedding of G on S. Euler's Formula says that Each degree-two vertex is on at least one face (but possibly at most one face as well), so F ≥ V 2 . Thus We may eliminate V 3 from this equation by noting that Dividing through by E produces the desired result. Notice that in Theorem 3.1 we subtract χ(S) from our lower bound. In effect, this increases the lower bound for embeddings on most surfaces, as χ(S) < 0 as long as S is not the projective plane, Klein bottle, torus, or sphere. Even when S is a torus or Klein bottle, the lower bound given in Theorem 3.1 is stronger than given by [1] (b). In the case of a closed 2-cell embedding, we have a result that gives a stronger bound than in [1] (b) and [1] (c) as long as χ(S) ≤ 0.