Triangle-free planar graphs with small independence number

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3−ε) for some fixed ε > 0. We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-free graph G′ such that α(G′) − |G′|/3 = α(G) − |G|/3 and |G′| ≤ (α(G) − |G|/3)/ε. We derive a number of algorithmic consequences as well as a structural description of n-vertex plane trianglefree graphs whose independence number is close to n/3. What is the smallest independence number a planar graph on n vertices can have? By Four Colour Theorem, each such graph is 4-colourable, and the largest colour class gives an independent set of size at least n/4. On the other hand, there are infinitely many planar graphs for that this bound is tight. In fact, it is an intriguing open problem to describe such graphs, and we do not even know any polynomial-time algorithm to decide whether an n-vertex planar graph has an independent set larger than n/4. In this paper, we study an easier related problem regarding independent sets in planar triangle-free graphs. By Grötzsch’ theorem [15], these graphs are 3colourable, and thus such a graph with n vertices has an independent set of size at least n/3. Unlike the general case, this bound is not tight—Steinberg and Tovey [19] proved the lower bound (n + 1)/3, and gave an infinite family ∗Computer Science Institute (CSI) of Charles University, Malostranské náměst́ı 25, 118 00 Prague, Czech Republic. E-mail: [email protected]. Supported by project GA1419503S (Graph coloring and structure) of Czech Science Foundation. †University of Warwick, Coventry CV4 7AL, UK. E-mail: [email protected]. The work of this author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648509). This publication reflects only its authors’ view; the European Research Council Executive Agency is not responsible for any use that may be made of the information it contains. 1 ar X iv :1 70 2. 02 88 8v 1 [ m at h. C O ] 9 F eb 2 01 7 of graphs for that this bound is tight. Dvořák et al. [6] improved this bound to (n + 2)/3 except for the graphs from this family. Furthermore, Dvořák and Mnich [7] proved that there exists ε > 0 such that each n-vertex plane trianglefree graph in that every 4-cycle bounds a face has an independent set of size at least n/(3− ε). Let us recall that α(G) denotes the size of the largest independent set in G. Dvořák and Mnich [7] moreover described an algorithm with time complexity 2 √ n that for an n-vertex planar triangle-free graph G and an integer a ≥ 0 decides whether α(G) ≥ (n + a)/3. This algorithm is based on a generalization of the last result of the previous paragraph, which we will state after introducing a couple of definitions. Let G be a plane graph and let H be a subgraph of G, whose drawing in the plane is inherited from G. Note that each vertex or edge of G that is not contained in H is drawn in some face of H; we say that the faces of H in that some part of G is drawn are full. Equivalently, a face f of H is full if and only if f is not a face of G. The supergraph of H with respect to that we define the fullness of a face of H will always be clear from the context. By |G|, we mean the number of vertices of G. A k-face of a plane graph is a face homeomorphic to an open disk bounded by a cycle of length k. We are now ready to state the result. Theorem 1 (Dvořák and Mnich [7]). There exists a constant γ > 0 as follows. Let G be a plane triangle-free graph and let H be its subgraph. If every 4-cycle in H bounds a face and every full face of H is a 4-face, then α(G) ≥ |G|+γ|H| 3 . Hence, if α(G) ≤ (n+ a)/3, then G contains no such subgraph H with more than a/γ vertices, and consequently it is easy to see that the tree-width of G is at most O( √ a). This gives the aforementioned algorithm by using the standard dynamic programming approach to deal with the bounded tree-width graph. Our main result is a more precise characterization of n-vertex plane trianglefree graphs with no independent set larger than (n + a)/3. To state the result, we need to give a few more definitions. We construct a sequence of graphs T1, T2, . . . , which we call Thomas-Walls graphs (Thomas and Walls [20] proved that they are exactly the 4-critical graphs that can be drawn in the Klein bottle without contractible cycles of length at most 4). Let T1 be equal to K4. For k ≥ 1, let u1u3 be any edge of Tk that belongs to two triangles and let Tk+1 be obtained from Tk − u1u3 by adding vertices x, y and z and edges u1x, u3y, u3z, xy, xz, and yz. The first few graphs of this sequence are drawn in Figure 1. For k ≥ 2, note that Tk contains unique 4-cycles C1 = u1u2u3u4 and C2 = v1v2v3v4 such that u1u3, v1v3 ∈ E(G). Let T ′ k = Tk − {u1u3, v1v3}. We also define T ′ 1 to be a 4-cycle C1 = C2 = u1v1u3v3. We call the graphs T ′ 1, T ′ 2, . . . reduced Thomas-Walls graphs, and we say that u1u3 and v1v3 are their interface pairs. Let us remark that the tight graphs found by Steinberg and Tovey [19] are precisely those obtained from