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Graphs with chromatic roots in the interval (1,2)

We present an infinite family of 3-connected non-bipartite graphs with chromatic roots in the interval (1,2) thus resolving a conjecture of Jackson's in the negative. In addition, we briefly consider other graph classes that are conjectured to have no chromatic roots in (1,2).


Introduction
The chromatic polynomial of a graph G is the function P (G, k) that counts the number of k-colourings of G when k is a natural number. It is well known that P (G, k) is a polynomial with degree equal to the number of vertices of G, and there is an extensive literature on the relationship between the properties of a graph and its chromatic polynomial. For general backgound information and references to much of this literature we refer the reader to the recent book on chromatic polynomials by Dong, Koh & Teo [2].
One aspect of the study of chromatic polynomials that has received considerable recent interest is the problem of the distribution of the chromatic roots of a graph -that is, the real and complex zeros of the chromatic polynomial. In a comprehensive survey paper, Jackson [4] describes much of this work and presents a number of open problems and conjectures. One of these conjectures arises from attempts to extend the portion of the real line on which the behaviour of the chromatic polynomial is completely understood. In particular, we have complete knowledge of the real roots of P (G, x) for all values of x ≤ 32/27, as described in the following fundamental theorem.
Theorem 1 (See Jackson [4]). If G is a loopless graph with n vertices, c components and b blocks which are not isolated vertices, then (a) P (G, x) is non-zero with sign (−1) n for x ∈ (−∞, 0); (b) P (G, x) has a zero of multiplicity c at x = 0; (c) P (G, x) is non-zero with sign (−1) n+c for x ∈ (0, 1); (d) P (G, x) has a zero of multiplicity b at x = 1; For general graphs, the constant 32/27 cannot be replaced by anything larger because Jackson [3] produced a family of graphs with real chromatic roots arbitrarily close to 32/27. However for certain classes of graphs the chromatic-root-free interval can be extendedfor example, Thomassen [6] showed that graphs with a hamiltonian path have no chromatic roots in (1, 1.29559 . . .). Both Jackson's graphs with chromatic roots close to 32/27 and Thomassen's graphs with chromatic roots close to 1.29559 have cutsets of size two, raising the question of whether the chromatic-root-free interval can be significantly extended for 3-connected graphs. Elementary continuity arguments show that bipartite graphs of odd order have a chromatic root in (1, 2) and these can have arbitrarily high connectivity, but Jackson conjectured that these were the only such examples.
In this paper we demonstrate that this conjecture is false, by providing an infinite family of 3-connected non-bipartite graphs with chromatic roots in (1,2), and then briefly discuss some alternative conjectures.

An infinite family of counterexamples
We consider a family of graphs whose smallest member is the 11-vertex graph X(3, 3) shown in Figure 1. It is clear by inspection that X(3, 3) is 3-connected and not bipartite and it is sufficiently small that its chromatic polynomial can be easily computed. The result of this computation is that which has real roots at r 1 ≈ 1.90263148 and r 2 ≈ 2.42196189. Therefore we conclude that Jackson's conjecture is false. Now let X(s, t) be the graph obtained by replacing the two independent sets of size three labelled S and T in Figure 1 by independent sets of size s ≥ 3 and t ≥ 3 respectively. Our aim is to prove the following theorem: to the chromatic polynomial.
It is straightforward, though tedious, to confirm that there are exactly 27 different types of colouring for these five vertices, and therefore the chromatic polynomial of X(s, t) is the sum of 27 terms of a similar form to the one shown in (1), although some of terms can be combined due to the symmetries of the situation. Overall however, the final expression is both complicated and unenlightening and more suitable for a symbolic algebra package such as Maple than for presentation in this paper. Fortunately however, we only need a small portion of the chromatic polynomial in order to deduce enough information for our purposes.
In particular, when s and t are both odd Proof. The derivative of P (X(s, t), x) can be obtained by differentiating each of the 27 terms in the chromatic polynomial separately. However as our concern is only with the value of this derivative at x = 2, we can ignore any terms with factors of (x − 2) s or (x − 2) t (or both) as these make no contribution to the final value (recall that s, t > 1). Therefore we need only consider the types of colouring that use either 1 or 3 colours on {v 0 , v 1 , v 3 } and similarly for {v 0 , v 2 , v 4 }. These colourings contribute the following five terms to the chromatic polynomial: t counts the colourings of the type shown in Figure 3.) The sum of the last two terms is and so these also make no net contribution to the derivative of P (X(s, t), x) evaluated at x = 2. Differentiating the remaining three terms and substituting x = 2 gives the stated result, and the second part follows immediately. When s and t are both odd, the graph X(s, t) has an odd number of vertices and as it is 3-connected, the chromatic root at x = 1 has multiplicity one. Therefore P (X(s, t), x) is negative on (−∞, 0), positive on (0, 1) and negative on some small interval (1, 1 + α). However by Lemma 4, the chromatic polynomial is decreasing as it passes through the root at x = 2 and so is positive on some small interval (2 − β, 2). It follows by continuity that the chromatic polynomial has a root in the interval (1, 2).
We can perform a similar analysis on the graphs obtained by adding the single edge v 1 v 4 to X(s, t) with the following result.

Questions and Conjectures
The results of the previous section raise two immediate questions: Question 6. What is the largest δ such that (1, δ) is a chromatic-root-free interval for 3connected non-bipartite graphs?
Question 7. Which classes of graphs have no chromatic roots in (1, 2)?
The first question is asking whether there are other 3-connected non-bipartite graphs with chromatic roots smaller than 1.90263148 and if so, how much smaller. The first obvious s\t   3  5  7  9  11  13  15  17  19  3 1.9026   Table 2: Chromatic root of Y (s, t) to 4 decimal places place to look is to see what happens when varying s and t for X(s, t) and Y (s, t). However the proof of our main theorem is an existence proof and gives us no clues as to the actual location of the chromatic roots other than that they are in (1,2). However computational evidence (detailed in Table 1 and Table 2) suggests that X(3, 3) contributes the smallest chromatic roots in these families.
A related question is obtained by not excluding bipartite graphs and just requiring 3connectivity. In this case, the smallest known chromatic root in (1, 2) is approximately 1.7811, which comes from the complete bipartite graph K 3,4 .
With respect to the second question, a number of different classes have been considered by various authors. As all (previously) known examples of 2-connected graphs with chromatic roots in (1, 2) had cutsets of size two or were bipartite of odd order, it was natural to conjecture that these were the only such graphs. However, with the considerable advantage of hindsight, the condition "3-connected and non-bipartite" is somewhat unsatisfactory because an ideal condition would intrinsically exclude odd order bipartite graphs rather than explicitly omit them from consideration.
One such condition was given by Thomassen [5], who conjectured that hamiltonian graphs do not have chromatic roots in (1,2). Recall that a graph G is said to be 1-tough if there is no subset S of its vertices such that G−S has more than |S| components, and that a hamiltonian graph is necessarily 1-tough. Now X(s, t) and Y (s, t) both have the property that they have sets of 3 vertices (e.g. {v 0 , v 1 , v 3 }) whose removal leaves more than 3 components, and therefore they are not 1-tough and hence not hamiltonian. Thus Thomassen's conjecture remains open.
In recent work, Dong & Koh [1] have found a number of sufficient conditions for graphs to have no chromatic roots in (1,2), such as the following useful result.
Theorem 8 (Dong & Koh [1]). If the graph G has a hamiltonian path v 1 ∼ v 2 ∼ · · · ∼ v n such that for all i > 2, the vertex v i has at least one neighbour in {v 1 , v 2 , . . . , v i−2 } (i.e. at least two "back neighbours" in total including v i−1 ) then G has no chromatic roots in (1, 2).
One feature common to all of the graphs that Dong & Koh can prove to be chromaticroot-free in the interval (1, 2) is that they satisfy a condition that is a weaker version of 1-toughness, namely that they have no independent sets S whose removal leaves more than |S| components. This suggests that maybe it is the toughness of hamiltonian graphs (rather than the actual cycle itself) that is important, and along these lines Dong & Koh have made the following interesting strengthening of Thomassen's conjecture.
Conjecture 9 (Dong & Koh [1]). If G is a graph with a chromatic root in (1, 2), then G has an independent set S such that c(G − S) > |S| where c(G − S) denotes the number of connected components of G − S.