The largest real zero of the chromatic polynomial

It is proved that if every subcontraction of a graph G contains a vertex with degree at most k, then the chromatic polynomial of G is positive throughout the interval (k, c~); Kk+l shows that this interval is the largest possible. It is conjectured that the largest real zero of the chromatic polynomial of a z-chromatic planar graph is always less than X. For Z = 2 and 3, constructions are given for maximal maximally-connected Z-chromatic planar graphs (i.e., 3-connected quadrangulations for ~ = 2 and 4-connected triangulations for Z = 3) whose chromatic polynomials have real zeros arbitrarily close to (but less than) X. O. I n t r o d u c t i o n For a (simple) graph G, let x(G), P(G,t) and rmax(G) denote the chromatic number of G, its chromatic polynomial, and the largest real zero of its chromatic polynomial, respectively. The real zeros of P(G, t) are then contained in the closed interval [0, rm~x(G)]. The largest integer zero of P(G,t ) is o f course equal to z ( G ) 1. In [4] I proved that if n is large enough compared with m, then P(Km,n,t) has real zeros arbitrarily close to all integers i satisfying 2<<,i<<,1⁄2m. Since Z(Km, n ) 1 -1, this shows that in general there is no upper bound for the largest real zero of P(G, t) in terms of its largest integer zero. However, in special cases there may be such an upper bound. Rather trivially, the chordal (or triangulated, or rigid-circuit) graphs have chromatic polynomials whose only zeros are integers, and so clearly rmax(G) = z ( G ) 1 for such a graph. Slightly less trivially, the chromatic polynomials o f all outerplanar graphs were characterized in [3]; apart from the zero at 1, every other zero of such a polynomial is o f the form 1 plus a root o f unity, and so again rmax(G) = x ( G ) 1. It is easy to find examples to show that this is not true for planar graphs in general. However, for planar graphs I conjecture that rmax(G) < z(G) (but see the Addendum at the end of this paper): * E-mail: [email protected]. 0012-365X/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S0012-365X(96)00277-4 142 D.R. WoodaH/Discrete Mathematics 172 (1997) 141-153 Conjecture. If G is a Z-chromatic planar graph, then P(G, t) is positive throughout the interval [Z, cx~). For Z = 4 this is a well-known conjecture, which (for example) follows from the Birkhoff-Lewis conjecture [1]; but I have not seen it before for Z < 4. For X = 1 it is trivial, since if G is 1-chromatic then P(G, t) has a unique zero, at 0. We shall see that the conjecture (if true) is best possible for ~ = 2 and 3; specifically, in Section Z below we shall see how to construct maximal maximally-connected x-chromatic planar graphs (i.e., 3-connected plane quadrangulations for X = 2 and 4-connected Eulerian plane triangulations for Z = 3) whose chromatic polynomials have real zeros arbitrarily close to (but less than) XBeraha and Kahane [6] construct plane triangulations (not 5-connected) whose chromatic polynomials have real zeros arbitrarily close to 4. The chromatic polynomials of the octahedron Ca +/£2 and the pentagonal double pyramid C5 + K 2 have zeros at 2.546602... and 2.677814 . . . . respectively. I conjectured in [4] that the chromatic polynomial of a plane triangulation is non-zero throughout the intervals (2,2.546602...) and (2.677814 .. . . 3). In [5] I proved the first of these conjectures and repeated the second. The examples constructed in Section 3 below, which include non-Eulerian as well as Eulerian 4-connected triangulations, show that the second conjecture is actually false. However, I still conjecture that it is true for 5-connected triangulations (and over a longer interval, since (75 +/~2 is not 5-connected).