From light edges to strong edge-colouring of 1-planar graphs

A strong edge-colouring of an undirected graph G is an edge-colouring where every two edges at distance at most 2 receive distinct colours. The strong chromatic index of G is the least number of colours in a strong edge-colouring of G . A conjecture of Erd˝os and Nešetˇril, stated back in the 80 ’s, asserts that every graph with maximum degree ∆ should have strong chromatic index at most roughly 1 . 25∆ 2 . Several works in the last decades have conﬁrmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly 4∆ , and even to smaller values under additional structural requirements. In this work, we initiate the study of the strong chromatic index of 1 -planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of 1 -planar graphs with maximum degree ∆ and strong chromatic index roughly 6∆ . As an upper bound, we prove that the strong chromatic index of a 1 -planar graph with maximum degree ∆ is at most roughly 24∆ (thus linear in ∆ ). The proof of this result is based on the existence of light edges in 1 -planar graphs with minimum degree at least 3 .


Introduction
Planar graphs are those graphs which can be drawn in the plane in such a way that no two edges cross.Colouring planar graphs has been one of the most active fields of graph theory, due in particular to the investigations that led to the well-known Four-Colour Theorem [1,2].Since then, whenever considering new graph problems, it generally makes sense wondering what happens for planar graphs.These graphs, however, are far from catching the structure of real-world graphs; for a given problem, one possible next direction can thus be to consider graph families that enclose planar ones.
One of the most natural generalizations of planar graphs is that of 1-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once.These graphs were first considered by Ringel [22], as he was investigating a possible generalization of the Four-Colour Theorem.Since then, many aspects of 1-planar graphs have been considered in the literature, including structural aspects, colouring aspects, topological aspects, and so on.We refer the interested reader to the recent survey by Kobourov, Liotta and Montecchiani on this topic [18].
Our goal in this work is to initiate the study of the strong chromatic index of 1-planar graphs.For a graph G, a strong edge-colouring of G is an edge-colouring where no two edges at distance at most 2 are assigned the same colour.To make it more precise, let us recall that two edges e, f are at distance 1 if they share an end, while e and f are at distance 2 if they are not at distance 1 and an end of e is adjacent to an end of f .A strong edge-colouring of G can thus also be regarded as an edge-partition of G into induced matchings, or as a proper vertex-colouring of the square of the line graph of G.The strong chromatic index of G, denoted χ s (G), is the least number of distinct colours assigned by a strong edge-colouring of G.
The notion of strong edge-colouring was first introduced by Fouquet and Jolivet [12].One of the leading conjectures in this field is that of Erdős and Nešetřil [9], stated back in the 1980's (when no confusion is possible, we here and further denote by ∆ the maximum degree of a given graph): Conjecture 1.1 is still wide open in general.It was verified for graphs with maximum degree ∆ = 3 by Andersen [3] and Horák, Qing and Trotter [16], while, already for every ∆ ≥ 4, it is not known whether the conjecture is true or not.To date, certainly the most investigated class of graphs is that of planar graphs, which were first considered by Faudree, Gyárfás, Schelp and Tuza [11].Using a nice combination (to be described in Section 3) of the Four-Colour Theorem and Vizing's Theorem, they proved that every planar graph G has strong chromatic index at most 4∆ + 4, while, for every ∆ ≥ 2, there exist planar graphs with maximum degree ∆ and strong chromatic index 4∆ − 4. Thus, roughly speaking, the maximum value of the strong chromatic index of a planar graph with maximum degree ∆ is of order 4∆.
Many works aimed at investigating conditions for the strong chromatic index of planar graphs to drop to roughly 3∆ and even 2∆.Such conditions notably involve the value of ∆, of the girth (i.e., length of a smallest cycle), and of the maximum average degree (i.e., density of a densest subgraph).See [5,7,13,15,19] for several works in that line.
In this work, we give first results towards understanding how the strong chromatic index of 1-planar graphs behaves.In Section 3, we establish that the maximum value of the strong chromatic index over all 1-planar graphs is of order at most roughly 24∆ (Corollary 3.4), while, for every ∆ ≥ 5, there exist 1planar graphs with maximum degree ∆ and strong chromatic index roughly 6∆ (Proposition 3.1).Although our upper bound is probably far from tight, it indicates that 1-planar graphs is yet another class of graphs for which the maximum strong chromatic index is linear in ∆, and not quadratic in ∆ as stated in the Erdős-Nešetřil bound from Conjecture 1.1.
The proof of our upper bound makes use of the presence, under some circumstances, of light edges in 1-planar graphs, which are edges whose ends' degree sum is somewhat small (i.e., bounded by a constant), in the following sense.By a k-vertex of a graph, we mean a vertex with degree k.By an (x, y)-edge of a graph, we mean an edge one of whose ends is an x-vertex and the other is a y-vertex.Light edges in 1-planar graphs were first studied by Fabrici and Madaras, who notably proved that 1-planar graphs are 7-degenerate, and 3-connected 1-planar graphs have (≤ 20, ≤ 20)-edges [10].Later on, Hudák and Šugerek [17] proved that every 1-planar graph G with δ(G) ≥ 4 has a (4, ≤ 13)-, (5, ≤ 9)-, (6, ≤ 8)or (7, 7)-edge.In the latter work, the authors also provided an optimal result regarding the existence of light edges in 1-planar graphs G with δ(G) ≥ 5. Some other results of this sort exist, see e.g.those in [18].However, by the time where the results of the current work were obtained, no such result on light edges in 1-planar graphs G with δ(G) ≥ 3 was known.To fill in this gap, we originally proved that 1-planar graphs with minimum degree at least 3 have (≤ 29, ≤ 29)-edges.Independently, however, other comparable and even stronger results of this sort have appeared in the literature.We explain this situation in Section 2.
Towards that conjecture, a result we proved is the following: After a round of the review process the current paper has been through, we were notified by an anonymous referee of the existence of two papers, namely [20] by Li, Hu, Wang and Wang, and [21] by Niu and Zhang, in which are provided results that are quite comparable to, and even much better than, Theorem 2.2.Namely, in [20] the authors proved that every 1-planar graph G with δ(G) ≥ 2 has an (x, y)-edge with x + y ≤ 29 or a 2-alternating cycle (i.e., a cycle v 0 , . . ., v 2m−1 v 0 where v 0 , v 2 , . . ., v 2m−2 are 2-vertices), while in [21]  Looking at the publication history of [20] and [21], it seems that our Theorem 2.2 was actually the first result of this sort to be publicly available online, in a previous version of the current paper [4].This is the main reason why we keep using Theorem 2.2 in the current paper, instead of using the better results.We however omit its proof, which the interested reader can still find in [4].
Let us finally mention that using the better results from [20] and [21] in place of Theorem 2.2 in the proof of Theorem 3.3 (and thus of Corollary 3.4) would improve the obtained bounds, but by a constant additive term only.
3 Application to strong edge-colouring of 1-planar graphs Using, in particular, Theorem 2.2, we study in this section strong edge-colourings of 1-planar graphs.

Lower bounds
Let ∆ ≥ 5, and let K ∆ be the graph obtained from K 6 by attaching ∆ − 5 new pendant vertices to every vertex.It can be observed that every two edges of K ∆ are at distance at most 2 apart.Furthermore, K ∆ is clearly 1-planar since K 6 is the biggest 1-planar complete graph (see e.g.[8]).Thus, for every ∆ ≥ 5 there are 1-planar graphs with maximum degree ∆ and strong chromatic index 6∆ − 15.
Actually, 1-planar graphs with maximum degree ∆ ≥ 5 and slightly larger strong chromatic index exist, as attested by the following construction, depicted in Figure 1.Start from a K 6 on vertex set {u 1 , . . ., u 6 }, and replace each of the edges u 1 u 2 , u 3 u 4 and u 5 u 6 by a complete bipartite graph K 2,∆−4 .Denote the resulting graph by K * ∆ .By construction of K * ∆ , all u i 's have degree ∆ (while the other vertices have degree 2), and K * ∆ is 1planar, as attested by the fact that K 6 is 1-planar (see Figure 1).Its strong chromatic index is deduced in the following proposition.Proposition 3.1.Every graph K * ∆ has strong chromatic index 6∆ − 12. Consequently, for every ∆ ≥ 5, there exist 1-planar graphs with maximum degree ∆ and strong chromatic index 6∆ − 12.
Proof: Every edge of K * ∆ is incident to at least one of the u i 's, while the u i 's, with the exception of the pairs {u 1 , u 2 }, {u 3 , u 4 } and {u 5 , u 6 }, are all adjacent.It is easy to see that every two edges of K * ∆ are at For smaller values of ∆, i.e., ∆ ∈ {3, 4}, some blown-up C 5 's are examples of 1-planar graphs with larger strong chromatic index (see Figure 2).The blown-up C 5 with maximum degree 3 is an example of a 1-planar graph with maximum degree 3 and strong chromatic index 10, which is the maximum possible value for the strong chromatic index of a graph with maximum degree 3 (as proved in [3,16]).The blown-up C 5 with maximum degree 4 is an example of a 1-planar graph with maximum degree 4 and strong chromatic index 20.While Erdős and Nešetřil have conjectured that this is the maximum strong chromatic index of a graph with maximum degree 4 (recall Conjecture 1.1), this has not been proved yet.We however know that the strong chromatic index of a graph with maximum degree 4 is at most 21, as recently proved by Huang, Santana and Yu [14].Thus, it might be that there exist 1-planar graphs with maximum degree 4 and strong chromatic index 21, in case the Erdős-Nešetřil Conjecture turned out to be false.

Upper bounds
The upper bound on the strong chromatic index of planar graphs with maximum degree ∆ in Theorem 1.2 relies on a nice combination of Vizing's Theorem and the Four-Colour Theorem.Let us recall that Vizing's Theorem [23] states that every graph with maximum degree ∆ has a proper ∆or (∆ + 1)-edge-colouring, i.e., a colouring (with ∆ or ∆ + 1 colours) of the edges where no two adjacent edges are assigned the same colour.The Four-Colour Theorem [1,2] states that every planar graph has a proper 4-vertex-colouring, i.e., a colouring of the vertices with four colours where no two adjacent vertices are assigned the same colour.
The proof of the upper bound in Theorem 1.2 goes as follows.Let G be a planar graph.By Vizing's Theorem, G admits a proper (∆ + 1)-edge-colouring φ.For every colour i assigned by φ, let us consider the i-graph M i being the graph of the i-coloured edges being at distance exactly 2 in G.More precisely, the vertices v e of M i are those edges e of G with colour i by φ, and two such vertices v e and v f are joined by an edge in M i if the edges e and f are at distance exactly 2 in G. Translating a planar drawing of G to one of M i , it is not complicated to convince oneself that each M i is planar.By the Four-Colour Theorem, each M i thus admits a proper 4-vertex-colouring ψ i .This yields a strong (4∆ + 4)-edge-colouring of G, where each edge e gets colour (φ(e), ψ φ(e) (v e )).
Unfortunately, mimicking the exact same proof for 1-planar graphs is not immediate.While Vizing's Theorem can of course be applied on a 1-planar graph and there does exist a 1-planar analogue of the Four-Colour Theorem, namely the Six-Colour Theorem (stating that every 1-planar graph has a proper 6vertex-colouring, as proved by Borodin [6]), it can however be noted that, when G is 1-planar, an i-graph M i might not be 1-planar itself.To overcome this issue and get our upper bound, we will instead consider proper edge-colourings avoiding certain patterns, that will ensure 1-planarity of every resulting i-graph M i .
In what follows, for a 1-planar graph G, a good edge-colouring will refer to an edge-colouring φ such that none of the following three configurations appears (see Figure 3).3. Three edges e, f and g receiving the same colour by φ, where e and f are at distance 2, joined by an edge g crossing g (Configuration C).
The fact that Configuration A is forbidden implies that a good edge-colouring is always proper.It also implies that, for every colour i assigned by φ, the i-graph M i is well defined.We now prove that the fact that Configurations B and C are forbidden implies that each graph M i is 1-planar.Lemma 3.2.Let G be a 1-planar graph, and φ be a good edge-colouring of G.For every colour i assigned by φ, the i-graph M i is 1-planar.
Proof: Consider a 1-planar embedding of G in the plane, and let us focus on the i-graph M i defined by φ for some assigned colour i.From the embedding of G, we can directly derive a corresponding embedding of M i , where each vertex of M i is "shaped" just as the associated edge in G, and every edge of M i , which results from any corresponding edge of G, is drawn in M i the same way as in G.Note that the fact that Configurations A and B are forbidden implies that, in the resulting embedding, no two vertices of M i overlap.The fact that Configuration C is forbidden implies that, in the embedding, no edge of M i goes "through" a vertex.Thus, vertices of M i are drawn in well separate locations, and the only crossing elements of M i are edges.Now, by the embedding above, we get that the edges of M i correspond to actual edges of G, embedded in the similar way in the plane.From this we directly get that M i cannot have an edge crossed more than once, as otherwise G would have one as well, a contradiction to the choice of its embedding.
We now prove an upper bound on the minimum number of colours in a good edge-colouring of a 1-planar graph.
Proof: To make our arguments work, we need to prove a stronger statement dealing with missing edges that could be involved in crossings.More precisely, we define a ghost triplet as an ordered triplet (u, v, xy) where: • u, v, x, y are four pairwise distinct vertices; • uv ∈ E(G) and xy ∈ E(G); • xy is not crossed; • the embedding of G can be extended directly to a 1-planar embedding of G + uv (i.e., all vertices and edges (different from uv) remain drawn the same) in such a way that uv and xy cross.
In what follows, to prove the existence of good edge-colourings of G, we focus on even more restricted edge-colourings.Namely, given a set T of ghost triplets of G where each edge xy is involved in only one triplet (u, v, xy) ∈ T , we prove that G admits what we call a µ-T -edge-colouring φ, which is a good µ-edge-colouring where the following bad configuration also does not appear.The proof is by induction on the number of vertices and edges of a 1-planar graph G.We also prove it by looking at G as a graph being a subgraph of a graph with maximum degree ∆.This notion of ∆ is important to keep track of the number of ghost triplets (u, v, xy) involving a given vertex u.In particular, below, the number of ghost triplets involving u will never exceed ∆ − d(u).The number of colours we use 1. Assume δ(G) ≥ 3. Since G is 1-planar, according to Theorem 2.2, it has a (≤ 29, ≤ 29)-edge uv.
We here consider G = G − uv, and T defined as mentioned earlier.In particular, we retain the 1-planar embedding Γ of G for G .Since G is smaller than G, it has a µ-T -edge-colouring φ by the induction hypothesis, which we wish extend to G and T , i.e., to uv.According to the arguments above, the worst-case scenario is when u and v have degree precisely 29 in G (or 28 in G ) and are each involved in ∆ − 29 ghost triplets, and uv is crossed by an edge xy where d G (x) = d G (y) = ∆.
In that case, we have n There are thus at most 3∆ + 54 colours forbidden for uv, and we can thus extend φ with an available colour.In all cases, we can thus extend φ to uv because we have a pool of µ colours while there are at most µ − 1 constraints around.This concludes the proof of Theorem 3.3.Proof: By Theorem 3.3, G has a good (max{3∆ + 55, 4∆ − 1})-edge-colouring φ.Now, by Lemma 3.2, for every colour i assigned by φ, the graph M i is 1-planar, and thus admits a proper 6-vertex-colouring ψ i .Every two adjacent edges of G are assigned different colours by φ, while, for every two edges at distance 2 being assigned colour i by φ, the two corresponding vertices in M i receive different colours by ψ i .Thus φ and the ψ i 's yield a strong (6 • max{3∆ + 55, 4∆ − 1})-edge-colouring of G.

Figure 2 :
Figure 2: Examples of 1-planar graphs with maximum degree ∆ ∈ {3, 4} and large strong chromatic index.distance at most 2 from each other.Consequently, no two edges of K * ∆ can receive the same colour by a strong edge-colouring, and thus χ s (K* ∆ ) = |E(K * ∆ )| = 6∆ − 12.For smaller values of ∆, i.e., ∆ ∈ {3, 4}, some blown-up C 5 's are examples of 1-planar graphs with larger strong chromatic index (see Figure2).The blown-up C 5 with maximum degree 3 is an example of a 1-planar graph with maximum degree 3 and strong chromatic index 10, which is the maximum possible value for the strong chromatic index of a graph with maximum degree 3 (as proved in[3,16]).The blown-up C 5 with maximum degree 4 is an example of a 1-planar graph with maximum degree 4 and strong chromatic index 20.While Erdős and Nešetřil have conjectured that this is the maximum strong chromatic index of a graph with maximum degree 4 (recall Conjecture 1.1), this has not been proved yet.We however know that the strong chromatic index of a graph with maximum degree 4 is at most 21, as recently proved by Huang, Santana and Yu[14].Thus, it might be that there exist 1-planar graphs with maximum degree 4 and strong chromatic index 21, in case the Erdős-Nešetřil Conjecture turned out to be false.

1 .
Two adjacent edges e and f receiving the same colour by φ (Configuration A).

2 .
Two crossing edges e and f receiving the same colour by φ (Configuration B).

Figure 3 :
Figure 3: Fordibben patterns in good edge-colourings.Red thick edges represent sets of edges that cannot all receive the same colour.

4 .
A ghost triplet (u, v, xy) ∈ T where an edge incident to u, an edge incident to v, and xy receive the same colour by φ (Configuration D).

2 .
Assume G has a 1-vertex u with unique neighbour v.We again consider G = G−uv, and T defined as previously.Let us consider a µ-T -edge-colouring φ of G .This time, because d G (u) = 1, we have n A,u = n C,u = 0.Then, the most constraints is when u is involved in ∆ − 1 ghost triplets, and when uv is crossed and v has degree ∆.In that very case,n A,u = n Cu = n D,x = n D,v = 0, n A,v = n C,v = n C,x = n D,u = ∆ − 1,and n B = 1.Thus, there are at most 4∆ − 3 colours forbidden for uv by φ, and we can thus extend φ with an available colour.3.Assume G has a 2-vertex u, and let v be any neighbour of u.Consider G , T and φ as before.Because d G (u) = 2, we have n A,u = n C,u = 1.Then, the most constraints is when u is involved in ∆ − 2 ghost triplets, and when uv is crossed and v has degree ∆.In such a case, we haven A,u = n B = n C,u = 1, n A,v = n C,v = n C,x = ∆ − 1, n D,u = ∆ − 2, and n D,x = n D,v = 0.Thus, there are at most 4∆ − 2 colours forbidden for uv by φ, and we can thus extend φ with an available colour.