The Extremal Function and Colin de Verdi`ere Graph Parameter Rose McCarty∗y School of Mathematics Georgia Institute of Technology Atlanta, GA, U.S.A. [email protected] Submitted: Aug 3, 2017; Accepted: Apr 26, 2018; Published: May 25, 2018 c The author. Released under the CC BY-ND license (International 4.0). Abstract The Colin de Verdi`ereparameter µ(G) is a minor-monotone graph parameter with connections to differential geometry. We study the conjecture that for every integer t, if G is a graph with at least t vertices and µ(G) 6 t, then jE(G)j 6 t+1 tjV (G)j − 2 . We observe a relation to the graph complement conjecture for the Colin de Verdi`ereparameter and prove the conjectured edge upper bound for graphs G such that either µ(G) 6 7, or µ(G) > jV (G)j − 6, or the complement of G is chordal, or G is chordal. Mathematics Subject Classifications: 05C35, 05C83, 05C10, 05C50 1 Introduction We consider only finite, simple graphs without loops. Let µ(G) denote the Colin de Verdi`ereparameter of a graph G introduced in [7] (cf. [8]). We give a formal definition of µ(G) in Section 2. The Colin de Verdi`ereparameter is minor-monotone; that is, if H is a minor of G, then µ(H) 6 µ(G). Particular interest in this parameter stems from the following characterizations: Theorem 1. For every graph G: 1. µ(G) 6 1 if and only if G is a subgraph of a path. 2. µ(G) 6 2 if and only if G is outerplanar. 3. µ(G) 6 3 if and only if G is planar. ∗Partially supported by NSF under Grant No. DMS-1202640. yNow at the Department of Combinatorics and Optimization, University of Waterloo. the electronic journal of combinatorics 25(2) (2018), #P2.32 1 4. µ(G) 6 4 if and only if G is linklessly embeddable. Items 1, 2, and 3 were shown by Colin de Verdi`erein [7]. Robertson, Seymour, and Thomas noted in [25] that µ(G) 6 4 implies that G has a linkless embedding due to their theorem that the Petersen family is the forbidden minor family for linkless embeddings [26]. The other direction for 4 is due to Lov´aszand Schrijver [18]. See the survey of van der Holst, Lov´asz,and Schrijver for a thorough introduction to the parameter [13]. There is also a relation between the Colin de Verdi`ereparameter and Hadwiger's conjecture that for every non-negative integer t, every graph with no Kt+1 minor is t- colorable. Let χ(G) denote the chromatic number of a graph G and let h(G) denote the Hadwiger number of G. That is, h(G) is the largest integer so that G has the complete graph Kh(G) as a minor. Then µ(Kh(G)) = h(G) − 1, and so µ(G) > µ(Kh(G)) = h(G) − 1 [13]. So if Hadwiger's conjecture is true, then for every graph G, χ(G) 6 µ(G) + 1. Colin de Verdi`ereconjectured that every graph satisfies χ(G) 6 µ(G) + 1 in [7]. For graphs with µ(G) 6 3, this statement is exactly the 4-Color Theorem [2,24]. One way to look for evidence for Hadwiger's conjecture is through considerations of average degree. In particular Mader showed that for every family of graphs F, there is an integer c so that if G is a graph with no graph in F as a minor, then jE(G)j 6 cjV (G)j [19]. It follows by induction on the number of vertices that every graph G with no graph in F as a minor is 2c + 1-colorable. In fact Mader showed that: Theorem 2. [20] For t 6 5, if G is a graph with h(G) 6 t + 1 and jV (G)j > t, then t+1 jE(G)j 6 tjV (G)j − 2 . However asymptotically, as noted by Kostochka [16] and Thomason [31], based on Bollob´aset al. [5]: + Theorem 3. [16,31] There exists a constant c 2 R suchp that for every positive integer t there exists a graph G with h(G) 6 t + 1 and jE(G)j > ct log tjV (G)j. Furthermore, Kostochka showed that the lower bound in Theorem 1.3 also serves as an upper bound [16]. This gives the best knownp bound on Hadwiger's conjecture, that graphs G with no Kt minor have χ(G) 6 O(t log t). We will study the following conjecture, that is an analog of Theorem2 for the Colin de Verdi`ereparameter: Conjecture 4. For every integer t, if G is a graph with µ(G) 6 t and jV (G)j > t, then t+1 jE(G)j 6 tjV (G)j − 2 . Nevo asked if this is true and showed that his Conjecture 1.5 in [22] implies Conjecture 4. Tait also asked this question as Problem 1 in [29] in relation to studying graphs with maximum spectral radius of their adjacency matrix, subject to having Colin de Verdi`ere parameter at most t. Butler and Young showed the following weakening of Conjecture4: Theorem 5. [6] For every integer t, if G is a graph on at least t vertices with zero t+1 forcing number no more than t, then jE(G)j 6 tjV (G)j − 2 . This bound is tight. the electronic journal of combinatorics 25(2) (2018), #P2.32 2 The zero forcing number is a graph parameter that is always at least the Colin de Verdi`ereparameter of a graph [1,3]. We observe that since the zero forcing number of a graph is also always at least the pathwidth of the graph [3], grids have unbounded zero forcing number and Colin de Verdi`ereparameter at most three. There are no known explicit constructions of graphs satisfying Theorem3. The essen- tial observation is instead that the Hadwiger number of Erd}os-R´enyi random graphs is too small. So it would be very interesting to know if random graphs are a counterexample to Conjecture4, and in particular to answer the following. Problem 6. What is the Colin de Verdi`ereparameter of the Erd}os-R´enyi random graph? Hall et al. studied this problem for some parameters related to the Colin de Verdi`ere parameter [10]. These related parameters are at least the vertex connectivity of a graph [Theorem 4, 12]. So random graphs do not give counterexamples to the analog of Con- jecture4 for these other parameters. Thus it seems that new techniques particular to the Colin de Verdi`ereparameter will be needed to solve this problem. We also observe that there is a relation between Conjecture4 and the graph comple- ment conjecture for the Colin de Verdi`ereparameter. Let G denote the complement of G. The graph complement conjecture for the Colin de Verdi`ereparameter is as follows: Conjecture 7. For every graph G, µ(G) + µ(G) > jV (G)j − 2. This conjecture was introduced by Kotlov, Lov´asz,and Vempala, who showed that the conjecture is true if G is planar [17]. Their result is used in this paper and will be stated formally in Section 4. Conjecture7 is also an instance of a Nordhaus-Gaddum sum problem. See the recent paper by Hogben for a survey of Nordhaus-Gaddum problems for the Colin de Verdi`ereand related parameters, including Conjecture7[11]. We observe that: + Observation 8. If there exists a constant c 2 R so that for every graph G, jE(G)j 6 + cµ(G)jV (G)j, then there exists a constant p 2 R so that for every graph G, µ(G)+µ(G) > pjV (G)j. This follows from noting that we would have cµ(G)jV (G)j + cµ(G)jV (G)j > jE(G)j + jV (G)j jE(G)j = 2 . So Conjecture4 would imply an asymptotic version of the graph com- plement conjecture for the Colin de Verdi`ereparameter. This weaker version is currently not known. In the other direction we will show in Section 2 that: Observation 9. If for every graph G, µ(G) + µ(G) > jV (G)j − 2, then every graph G µ(G)+2 has jE(G)j 6 (µ(G) + 1)jV (G)j − 2 . Then in particular the graph complement conjecture for Colin de Verdi`ereparameter would imply that all graphs G are 2µ(G)+2-colorable. We next comment on the tightness of Conjecture4. We say that a graph G is the join of non-empty graphs H1 and H2 if the vertex set of G is the disjoint union of V (H1) and V (H2), and for i = 1; 2 the induced subgraph of G on vertex set V (Hi) is the graph Hi, and for every pair of vertices u 2 V (H1) and v 2 V (H2), uv 2 E(G). We will show in Section 2 that: the electronic journal of combinatorics 25(2) (2018), #P2.32 3 Observation 10. Let H be any edge-maximal planar graph on at least four vertices and let t > 3 be an integer. Let G denote the join of H and Kt−3. Then µ(G) = t and t+1 jE(G)j = tjV (G)j − 2 . So for every positive integer t, Conjecture4 is tight for infinitely many graphs. We say a graph G is chordal if for every cycle C of G of length greater than 3, the induced subgraph of G with vertex set V (C) has some edge that is not in E(C).