On the Strong Chromatic Index of Sparse Graphs

The strong chromatic index of a graph $G$, denoted $\chi_s'(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi_{s,\ell}'(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with $\operatorname{girth}(G) \geq 41$ then $\chi_{s,\ell}'(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and $\operatorname{girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper. Finally, if $G$ is a planar graph with maximum degree at most four and $\operatorname{girth}(G) \geq 28$, then $\chi_s'(G) \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.


Introduction
A proper edge-coloring of a graph G is an assignment of colors to the edges so that incident edges receive distinct colors. A strong edge-coloring of a graph G is an assignment of colors to the edges so that edges at distance at most two receive distinct colors. A proper edge-coloring is a decomposition of G into matchings, while a strong edge-coloring is a decomposition of G into induced matchings. Fouquet and Jolivet [10,11] defined the strong chromatic index of a graph G, denoted χ ′ s (G), as the minimum integer k such that G has a strong edge-coloring using k colors. Erdős and Nešetřil gave the following conjecture, which is still open, and provided an example to show that it would be sharp, if true.
Conjecture 1.1 (Erdős and Nešetřil [8]). For every graph G, χ ′ s (G) ≤ Towards this conjecture, Molloy and Reed [16] bounded χ ′ s (G) away from the trivial upper bound of 2∆(G)(∆(G) − 1) + 1 by showing that every graph G with sufficiently large maximum degree satisfies χ ′ s (G) ≤ 1.998∆(G) 2 . Bruhn and Joos [5] have announced an improvement, claiming The focus of this paper is the study of strong edge-colorings of subcubic graphs, those with maximum degree at most three, and subquartic graphs, those with maximum degree at most four. Faudree, Gyárfas, Schelp, and Tuza [9] studied χ ′ s (G) in the class of subcubic graphs, and gave the following conjectures. Conjecture 1.2 (Faudree et al. [9]). Let G be a subcubic graph.
Several papers prove sharper bounds on the strong chromatic index of planar graphs with additional structure [11,12,13,14], generally by introducing conditions on maximum average degree or girth to ensure that the target graph is sufficiently sparse. For a graph G, the maximum average degree of G, denoted mad(G), is the maximum of average degrees over all subgraphs of G. Hocquard, Montassier, Raspaud, and Valicov [12,13] proved the following.    [13]). Let G be a subcubic graph.
Parts (1) and (2) of Theorem 1.3 are sharp by the graphs shown in Figures 2 and 3, respectively. An elementary application of Euler's Formula (see [20]) gives the following. Proposition 1.4. If G is a planar graph with girth g then mad(G) < 2g g−2 . Theorem 1.3 and Proposition 1.4 yield the following corollary. Corollary 1.5 (Hocquard et al. [13]). Let G be a subcubic planar graph with girth g.
Note that no non-trivial sparsity condition on a graph G with maximum degree d will guarantee that χ ′ s (G) < 2d − 1 since any graph having two adjacent vertices of degree d requires at least 2d − 1 colors to strongly edge-color the graph. We give sparsity conditions that imply a subcubic planar graph has strong chromatic index at most five and a subquartic planar graph has strong chromatic index at most seven. Previous work in this direction was initiated by Borodin and Ivanova [3], Chang, Montassier, Pěcher, and Raspaud [6], and most recently extended by Wang and Zhao [19]. The current-best bounds are given by the following two results. Theorem 1.6 (Borodin and Ivanova [3]). Let G be a subcubic graph.
One barrier to proving sparsity conditions that imply χ ′ s (G) ≤ 5 is that there exist graphs G with mad(G) = 2 and χ ′ s (G) = 6. Let S 3 be a triangle with pendant edges at each vertex, and let S 4 be a 4-cycle with pendant edges at two adjacent vertices. For k ≥ 5, let S k be a k-cycle with pendant edges at each vertex. Each of S 3 , S 4 and S 7 have maximum average degree 2 and strong chromatic index at least 6, see Figure 4. However, these graphs are 6-critical with respect to χ ′ s (G), as the removal of any edge from S 3 , S 4 or S 7 results in a graph that has a strong edge-coloring using five colors. Our main theorem demonstrates that if these few graphs are avoided, and the maximum average degree is not too large, then we can find a strong 5-edge-coloring, improving Theorem 1.6. Theorem 1.8. Let G be a subcubic graph.
Using similar methods, we improve the bounds in Theorem 1.7 when d = 4. Theorem 1.9. Let G be a subquartic graph.
We also consider a list variation of the strong chromatic index of G, first introduced by Vu [21]. A strong list edge-coloring of a graph G is an assignment of lists to E(G) such that a strong edgecoloring can be chosen from the lists at each edge. The minimum k such that a graph G can be strongly list edge-colored using any lists of size at least k on each edge is the strong list chromatic index of G, denoted χ ′ ℓ,s (G). Borodin and Ivanova [3] asked if there are sparsity conditions that imply χ ′ ℓ,s (G) ≤ 2d − 1 for a planar graph G with maximum degree d. We generalize the bounds in Theorem 1.6 to apply to list coloring. Theorem 1.10. Let G be a subcubic graph.
The proofs of Theorems 1.8, 1.9, and 1.10 use the discharging method. We begin by proving Theorem 1.10 in Section 2 as the proof is shorter and the one reducible configuration is used again in the proof of Theorem 1.8 in Section 3.

Preliminaries and Notation
Throughout this paper we will only consider simple, finite, undirected graphs. We refer to [20] for any undefined definitions and notation. A graph G has vertex set V (G), edge set E(G), and maximum degree ∆(G).
If a vertex v has degree j we refer to it as a j-vertex, and if v has a neighbor that is a j-vertex, we say it is a j-neighbor of v. When G is planar we let F (G) denote the set of faces of G, and ℓ(f ) denote the length of a face f . The girth of a graph G is length of its shortest cycle. A graph G is Additionally, let the contracted graph of G, denoted ct(G) be the graph obtained by deleting all Thus, for the remainder of the paper a vertex v is a k + -vertex in G if it has degree at least k in ct(G).
We will make use of the discharging method for some of our results. For an introduction to this method, see the survey by Cranston and West [7]. We will directly use two standard results that can be proven using this method. Both of Theorems 1.6 and 1.7 rely on Lemmas 1.11 and 1.12.
Let G be a graph and ct(G) be the contracted graph. An ℓ-thread is a path Lemma 1.11 (Cranston and West [7]). If G is a graph with girth at least ℓ + 1 and mad(G) < 2 + 2 3ℓ−1 , then ct(G) contains a 1-vertex or an ℓ-thread.

Strong List Edge-Coloring of Subcubic Graphs
In this section, we prove Theorem 1.10. Our proof uses the discharging method, wherein we assign an initial charge to the vertices and faces of a theoretical minimal counterexample. This initial charge is then disbursed according to a set of discharging rules in order to draw a contradiction to the existence of such a minimal counterexample. We will often make use of the following, which is another simple and well known application of Euler's Formula.
We will also use the Combinatorial Nullstellensatz, which will be applied to show we can extend certain list colorings.
Theorem 2.2 (Combinatorial Nullstellensatz [1]). Let f be a polynomial of degree t in m variables over a field F. If there is a monomial The first item of Theorem 1.10 follows from the following strengthened theorem. Proof. For the sake of contradiction, select G, p, c, and L as in the theorem statement, and assume there does not exist a strong edge coloring of E(G) using colors from L. In this selection, minimize n(G). Note that G is connected and e(G) > 5. We can further assume that d(p) > 1, since if d(p) = 1 and {p ′ } = N (p) then we can instead color the edges incident to p ′ .
Proof. Suppose that G contains a cut-edge uv with d(u) = d(v) = 3. There are exactly two components in G − uv, call them G 1 and G 2 , with u ∈ V (G 1 ) and v ∈ V (G 2 ). Without loss of generality, Thus there is a strong edge-coloring of G ′ 1 using the 5-listassignment L. Next, color the other two edges incident to v using colors distinct from those on the edges incident to u. Now, G ′ 2 is a subcubic planar graph of girth at least 41 with distinctly colored edges about the vertex v and n(G ′ 2 ) < n(G). Thus, there is an extension of the coloring to G ′ 2 . The colorings of G ′ 1 and G ′ 2 form a strong edge coloring of G, a contradiction.
By removing all edges incident to v 2 , . . . , v 7 and u 1 , . . . , u 8 , as well as any isolated vertices that are produced, we obtain a graph G ′ with fewer vertices than G, so we can strongly edge-color G ′ with 5 colors. We fix such a coloring of G ′ and generate a contradiction by extending this coloring to a strong edge-coloring of G. Suppose that c 1 , . . . , c 6 are the colors of the edges incident to the vertices v 0 and v 9 , and assign variables y 1 , . . . , y 8 to the pendant edges, and variables x 1 , . . . , x 7 to the interior edges as shown in Figure 7. Identifying the conflicts between variables and colors produces the following polynomial, f (y 1 , . . . , y 8 , x 1 , . . . , We will use the Combinatorial Nullstellensatz to show that there is an assignment of colorŝ c 1 , . . . ,ĉ 8 and c ′ 1 , . . . , c ′ 7 such that f (ĉ 1 , . . . ,ĉ 8 , c ′ 1 , . . . , c ′ 7 ) = 0. Such an assignment of colors would extend the inductive coloring of G − p to a strong edge-coloring of G. If the coefficient of (x 1 x 2 x 3 x 4 x 5 x 6 x 7 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 ) 4 is nonzero, then there are values from L for x 1 , . . . , x 7 , y 1 , . . . , y 8 such that f is nonzero by Theorem 2.2. Using the Magma algebra system [4], this monomial has coefficient −2, and thus there is a strong edge-coloring using the 5-list assignment 1 . Thus, the 8-caterpillar does not exist in a vertex minimal counterexample.
Note that the proof in Lemma 2.5 cannot be extended to exclude a 7-caterpillar in G, as there exists a 5-coloring of the external edges that does not extend to the caterpillar, even when the lists are all the same.
To complete the proof, we apply a discharging argument to ct(G). 2 . First, observe that by Lemma 2.4, ct(G) is 2-connected and so every face is a simple cycle of length at least 41. Also observe that by Lemma 2.5, ct(G) does not contain a path of length 8 where every vertex is of degree 2, unless one of those vertices is p.
Assign charge 2d(v) − 6 to every vertex v = p, charge ℓ(f ) − 6 to every face f , and charge 2d(p) + 5 to p. By Proposition 2.1, the total amount of charge on ct(G) is −1. Apply the following discharging rules.
(R1) For every v ∈ G − p, if v is a 2-vertex, v pulls charge 1 from each incident face.
(R2) If p is a 2-vertex, then p gives charge 9 2 to each incident face.
Observe that every vertex has nonnegative charge after this discharging process. It remains to show that every face has nonnegative charge.
Let f be a face, and let r 2 be the number of 2-vertices on the boundary of f , not counting p, and consider two cases.
Case 1: d(p) = 3 or p is not adjacent to f . In this case, p does not give charge to f , and therefore f has charge ℓ(f )−r 2 −6 after discharging. Also, the boundary of f does not contain a path of length 8 containing only vertices of degree 2, thus Case 2: d(p) = 2 and p is adjacent to f . By (R2), p gives charge 9 2 to f , so that f has charge ℓ(f ) − r 2 − 3 2 after discharging. The boundary of f does not contain a path of length 8 containing only vertices of degree 2, except when using p, so, r 2 ≤ 7 8 ℓ(f ) . Since ℓ(f ) ≥ 41, we have Thus, all vertices and faces have nonnegative charge, contradicting Proposition 2.1.
The second item of Theorem 1.10 follows by similarly strengthening the statement to include a precolored vertex and using Lemmas 1.11, 2.4, and 2.5.

Strong Edge-Coloring of Sparse Graphs
In this section, we prove Theorems 1.8 and 1.9.
Let G be a graph with maximum degree ∆(G) ≤ d. For a vertex v in ct(G) denote by N 3 (v) the set of 3 + -vertices u where ct(G) contains a path P from u to v where all internal vertices of P are 2 ⊥ -vertices. For u ∈ N 3 (v), let µ(v, u) be the number of paths from v to u whose internal vertices have degree 2 in ct(G). For a 3-vertex v, let the responsibility set, denoted Resp(v), be the set of 2 ⊥ -vertices that appear on the paths between v and the vertices in N 3 (v).
Let D be a subgraph of G. We call D a k-reducible configuration if there exists a subgraph D ′ of D such that any strong k-edge-coloring of G − D ′ can be extended to a strong k-edge-coloring of G. One necessary property for the selection of D ′ is that no two edges that remain in G − D ′ can have distance at most two in G but distance strictly larger than two in G − D ′ . In the next subsection we describe several reducible configurations.

Reducible Configurations
This subsection contains description of four types of reducible configurations. Each configuration is described in terms of how it appears within ct(G) where G is a graph with maximum degree ∆(G) ≤ d for some d ≥ 4.
Let t be a positive integer. The t-caterpillar is formed by two 3 + -vertices v 0 and v t+1 with a Let t 1 , . . . , t k be nonnegative integers. A configuration Y (t 1 , . . . , t k ) is formed by a k + -vertex v and k internally disjoint paths of lengths t 1 + 1, . . . , t k + 1 with v as a common endpoint, where the internal vertices of the paths are 2 ⊥ -vertices. We call such configuration a Y -type configuration about v, see Figure 8.
A configuration H(t 1 , t 2 ; r; s 1 , s 2 ) is formed by two 3-vertices u and v and 5 internally disjoint paths of lengths t 1 + 1, t 2 + 1, r + 1, s 1 + 1, and s 2 + 1, where the internal vertices of the paths are 2 ⊥ -vertices. The paths of lengths t 1 + 1 and t 2 + 1 have v as an endpoint, the path of length r + 1 has u and v as endpoints and the paths of lengths s 1 + 1 and s 2 + 1 have u as an endpoint. We call such configuration an H-type configuration about v and u, see Figure 9.
A configuration Φ(t, a 1 , a 2 , s) is formed by two 3-vertices u and v and 4 internally disjoint paths of lengths t + 1, a 1 + 1, a 2 + 1, and s + 1, where the internal vertices of the paths are 2 ⊥ -vertices. The path of length t + 1 has v as an endpoint, the paths of lengths a 1 + 1 and a 2 + 1 have u and v as endpoints and the path of length s + 1 has u as an endpoint. We call such configuration a Φ-type configuration about v and u, see Figure 10.
The reducibility of these configurations was verified using computer 3 , and in addition the 8caterpillar is addressed in Lemma 2.5. Given the definition of a 2 ⊥ -vertex, the vertices of degree two in these configurations may, or may not, be adjacent to some 1-vertices in G. We demonstrate the reducibility of the instances of these configurations wherein each vertex of degree 2 is adjacent to d − 2 1-vertices, as depicted in Figures 8-10. This suffices to address all other instances of these configurations that may occur.  [19]) For d ≥ 4, the (2d − 2)-caterpillar is (2d − 1)-reducible.

(Wang and Zhao
These caterpillars are likely the smallest that are reducible for each degree d. Thus, the bounds in Theorems 1.6 and 1.7 are best possible using only Lemma 1.12. To improve these bounds, we demonstrate larger reducible configurations and use a more complicated discharging argument.
Proof. Suppose that ct(G) contains a cut-edge uv. In G, the vertices u and v have degree at least two. There are exactly two components, G 1 and G 2 , in G − uv, with u ∈ V (G 1 ) and v ∈ V (G 2 ). Let u 1 , u 2 be neighbors of u in G 1 and v 1 , v 2 be neighbors of v in G 2 ; let u 1 = u 2 only when u has a unique neighbor in G 1 , and v 1 = v 2 only when v has a unique neighbor in G 2 . Let Since n(G ′ ) < n(G) and mad(G ′ ) ≤ mad(G), there is a strong 5-edge-coloring c of G ′ . Extend the coloring c to color c(vv 1 ) and c(vv 2 ) from the colors not in {c(uv), c(uu 1 ), c(uu 2 )}, a contradiction. We similarly reach a contradiction when G ′ 2 = G. Therefore, n(G ′ i ) < n(G) and mad(G ′ i ) ≤ mad(G) for each i ∈ {1, 2}. Thus, there exist strong 5-edge-colorings c 1 and c 2 of G ′ 1 and G ′ 2 , respectively. For each coloring, the colors on the edges uv, uu 1 , uu 2 , vv 1 , vv 2 are distinct. Let π be a permutation of the five colors satisfying π(c 2 (e)) = c 1 (e) for each edge e ∈ {uv, uu 1 , uu 2 , vv 1 , vv 2 }. Then, we extend the coloring c 1 of G ′ 1 to all of G by assigning c 1 (e) = π(c 2 (e)) for all edges e ∈ E(G ′ 2 ). The coloring c 1 is a strong 5-edge-coloring of G, a contradiction.
If ct(G) does not have any 3-vertices, then ct(G) must be isomorphic to cycle C n . If n ≥ 9, then ex 3 (G) contains an 8-caterpillar. If n ∈ {5, 6, 8}, then G is a subgraph of S 5 , S 6 , or S 8 , which each has a strong edge-coloring using five colors, discovered by computer. When n ∈ {3, 4, 7}, G does not contain S 3 , S 4 , or S 7 , and any proper subgraph of these graphs is 5 strong edge-colorable, discovered by computer. Therefore, ct(G) is not isomorphic to a cycle, and hence for every 2 ⊥ -vertex u in G, If G has some vertex v such that |N 3 (v)| = 1, then G must be a subgraph of Θ(t 1 , t 2 , t 3 ), which is the graph consisting of three internally disjoint x − y paths of length t 1 + 1, t 2 + 1 and t 3 + 1, for some 0 ≤ t 1 ≤ t 2 ≤ t 3 .
In all cases, we verified that the final charge is at least 2 + 1 7 , contradicting that the average degree of ct(G) is strictly less than 2 + 1 7 . We note that it is possible to improve the bound mad(G) < 2 + 1 7 by a small amount. In particular, the discharging method used above essentially states that the average size of a responsibility set in ct(G) is at most 12. By careful analysis, we can find that a 3-vertex v with | Resp(v)| ≤ 11 has some excess charge after the discharging argument that could be used to increase the charge on nearby vertices by a small fraction. We have verified using computation that for every 3-vertex v, there is at least one vertex u ∈ N 3 (v) where | Resp(u)| < 12. Thus, it is impossible to have a minimal counterexample where all responsibility sets have size 12, and it is feasible to construct a discharging argument that will improve on the bound mad(G) < 2 + 1 7 by a small fraction. We do not do this explicitly as it requires significant detail without significant gain.
In order to prove that mad(G) < 2 + 1 6 implies that G can be strongly 5-edge-colored, then the proof will imply that the average size of a responsibility set is at most 10. This will require sending charge to all of the vertices with 11 or 12 vertices in the responsibility set, and also making sure that the charge comes from vertices with responsibility sets much smaller. Likely, larger reducible configurations will grant some improvement in this direction, but our algorithm is insufficient to effectively test reducibility for larger configurations.

Proof of Theorem 1.9
Proof. Note that the second item of Theorem 1.9 follows from the first by Proposition 1.4. For the first item, we follow a similar discharging argument as in Theorem 1.8. The argument will be simpler as we will only discharge from 3 + -vertices to 2 ⊥ -vertices. Select a graph G that satisfies the hypotheses and minimizes n(G). Observe that ct(G) is 2-connected by an argument similar to Lemma 3.4.
Since the 6-caterpillar is 7-reducible by Claim 3.1, ct(G) does not contain a path of six 2 ⊥vertices. Since G has girth at least 7, ct(G) is not a cycle, so it contains at least one 3 + -vertex.
If v is a 3 + -vertex, then let Resp(v) be the set of 2 ⊥ -vertices reachable from v using only 2 ⊥vertices. We consider Resp(v) to be a multiset, where the multiplicity of a vertex u ∈ Resp(v) is given by the number of paths from v to u using only 2 ⊥ -vertices. Note that the multiplicity is either 1 or 2.