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An Optimal $\Chi $-Bound for ($ P 6 $, Diamond)-Free Graphs

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An Optimal $\Chi $-Bound for ($ P 6 $, Diamond)-Free Graphs An Optimal χ-Bound for (P6, diamond)-Free Graphs Kathie Cameron ∗ Shenwei Huang † Owen Merkel ‡ August 29, 2018 Abstract Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. Let Pt be the path on t vertices and Kt be the complete graph on t vertices. The diamond is the graph obtained from K4 by removing an edge. In this paper we show that every (P6, diamond)-free graph G satisfies χ(G) ≤ ω(G) + 3, where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. Our bound is attained by the complement of the famous 27-vertex Schl¨afli graph. Our result unifies previously known results on the existence of linear χ-binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect (P6, diamond)-free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well-known characterization of 3-colourable (P6,K3)-free graphs. 1 Introduction All graphs in this paper are finite and simple. We say that a graph G contains a graph H if H is isomorphic to an induced subgraph of G. A graph G is H-free if it does not contain H. For a family H of graphs, G is H-free if G is H-free for every H ∈ H. When H consists of two graphs, we write (H1,H2)-free instead of {H1,H2}-free. As usual, Pt and Cs denote the path on t vertices and the cycle on s vertices, respectively. The complete graph on n vertices is denoted by Kn. The graph K3 is also referred to as the triangle. Let the diamond be the graph obtained from K4 by removing an edge. For two graphs G and H, we use G + H to denote the disjoint union of G and H. For a positive integer r, we use rG to denote the disjoint union of r copies of G. The complement of G is denoted by G. A clique (resp. stable set) in a graph is a set of pairwise adjacent (resp. non-adjacent) vertices. A q-colouring of a graph arXiv:1809.00739v1 [math.CO] 3 Sep 2018 G is a function φ : V (G) −→ {1,...,q} such that φ(u) 6= φ(v) whenever u and v are adjacent in G. Equivalently, a q-colouring of G is a partition of V (G) into q stable sets. A graph is q-colourable if it admits a q-colouring. The chromatic number of a graph G, denoted by χ(G), is the minimum number q for which G is q-colourable. The clique number of G, denoted by ω(G), is the size of a largest clique in G. Obviously, χ(G) ≥ ω(G) for any graph G. ∗Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada, N2L 3C5. Email: [email protected]. Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06517. †Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada, N2L 3C5. Email: [email protected]. Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06517. ‡Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada, N2L 3C5. Email: [email protected]. Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06517 and an NSERC Undergraduate Student Research Award. 1 A family G of graphs is said to be χ-bounded if there exists a function f such that for every graph G ∈ G, every induced subgraph H of G satisfies χ(H) ≤ f(ω(H)). The function f is called a χ-binding function for G. The notion of χ-bounded families was introduced by Gy´arf´as [11] in 1987. Since then it has received considerable attention for H-free graphs. We briefly review some results in this area. A hole in a graph is an induced cycle of length at least 4. An antihole is the complement of a hole. A hole or antihole is odd or even if it is of odd or even length, respectively. The famous Strong Perfect Graph Theorem [6] says that the class of graphs without odd holes or odd antiholes is χ-bounded and the χ-binding function is the identity function f(x) = x. If we only forbid odd holes, then the resulting class remains χ-bounded but the best known χ-binding function is double exponential [17]. On the other hand, if even holes are forbidden, then a linear χ-binding function exists [1]: every even-hole- free graph G satisfies χ(G) ≤ 2ω(G) − 1. In recent years, there has been an ongoing project led by Scott and Seymour that aims to determine the existence of χ-binding functions for classes of graphs without holes of various lengths. We refer the reader to the recent survey by Scott and Seymour [18] for various nice results. One thing to note is that most χ-binding functions in this setting are exponential. Another line of research is the study of H-free graphs for a fixed graph H. A classsical result of Erd˝os [8] shows that the class of H-free graphs is not χ-bounded if H contains a cycle. Gy´arf´as [10] conjectured that the converse is also true (known as the Gy´arf´as Conjecture), and ω(G)−1 proved the conjecture when H = Pt [11]: every Pt-free graph G has χ(G) ≤ (t − 1) . Similar to results in [18], this χ-binding function is exponential in ω(G). It is natural to ask the following question. • Is it possible to improve the exponential bound for Pt-free graphs to a polynomial bound? This turns out to be a very difficult question, and not much progress has been made over the past 30 years. It remains open whenever t ≥ 5. (For t ≤ 4, Pt-free graphs are perfect and hence f(x) = x is the χ-binding function.) Therefore, researchers have started to investigate subclasses of Pt-free graphs, hoping to discover techniques and methods that would be useful for tackling the problem. A natural type of subclass is to forbid a second graph in addition to forbidding Pt. For example, it was shown by Gaspers and Huang [9] that every (P6,C4)- 3 free graph G has χ(G) ≤ 2 ω(G). This 3/2 bound was improved recently by Karthick and 5 Maffray [13] to the optimal bound 5/4: every (P6,C4)-free graph G has χ(G) ≤ 4 ω(G). In another work, Karthick and Maffray [14] showed that every (P5, diamond)-free graph G satisfies χ(G) ≤ ω(G) + 1. Bharathi and Choudum [2] gave a cubic χ-binding function for the class of (P2 + P3, diamond)-free graphs. For the class of (P6, diamond)-free graphs, a common superclass of (P5, diamond)-free graphs and (P2 + P3, diamond)-free graphs, Karthick and Mishra [15] proved that f(x) = 2x + 5 is a χ-binding function, greatly improving the result for (P2 + P3, diamond)-free graphs. In the same paper, they also obtained an optimal χ-bound for (P6, diamond)-free graphs when the clique number is 3: every (P6, diamond, K4)-free graph is 6-colourable. For more results of this flavor, see [4, 5, 12, 19]. Our Contributions In this paper, we give an optimal χ-bound for the class of (P6, diamond)-free graphs. We prove that each (P6, diamond)-free graph G satisfies χ(G) ≤ ω(G) + 3 (Theorem 3 in Section 4). The bound is tight since it is attained by the complement of the famous 27-vertex Schl¨afli graph [15]. Our result unifies the results on the existence of linear χ-binding functions for the class of (P5, diamond)-free graphs [14], (P2 + P3, diamond)-free graphs [2], and (P6, diamond)-free graphs [15], and answers an open question in [15]. The remainder of the paper is organized as follows. We present some preliminaries in 2 Section 2 and prove some structural properties of imperfect (P6, diamond)-free graphs in Section 3. We prove our main result in Section 4 and give some open problems in Section 5. 2 Preliminaries For general graph theory notation we follow [3]. Let G = (V, E) be a graph. The neighbourhood of a vertex v, denoted by NG(v), is the set of vertices adjacent to v. For a set X ⊆ V (G), let NG(X) = Sv∈X NG(v) \ X and NG[X] = NG(X) ∪ X. The degree of v, denoted by dG(v), is equal to |NG(v)|. For x ∈ V and S ⊆ V , we denote by NS(x) the set of neighbours of x that are in S, i.e., NS(x)= NG(x) ∩ S. For X,Y ⊆ V , we say that X is complete (resp. anti-complete) to Y if every vertex in X is adjacent (resp. non-adjacent) to every vertex in Y . For x ∈ V and Y ⊆ V , we say x is complete (resp. anti-complete) to Y if x is adjacent (resp. non-adjacent) to every vertex in Y . A vertex subset K ⊆ V is a clique cutset if G − K has more components than G and K induces a clique. For S ⊆ V , the subgraph induced by S is denoted by G[S]. We shall often write S for G[S] if the context is clear.
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