On Hadwigers Number of a graph with partial information

The celebrated Hadwiger's conjecture [1] relates two important parameters of a graph, the chromatic number χ(G) and h(G), the size of the largest clique minor of the graph. Specifically, it asserts that χ(G) ≤ h(G) (or, equivalently, that any l-chromatic graph has K l as a minor). The conjecture has been proved in several special cases (e.g. [2]), and for random graphs [3]. Thus it is of interest to compare and prove various inequalities involving one of the two parameters. This is described by: Definition 1 Let χ(G) ≤ w(G) be an inequality, valid for some graph parameter w(G) and all graphs G from a certain class C. We say that h(G) interpolates in the previous inequality iff relation h(G) ≤ w(G) is valid for all graphs G ∈ C. Of course, h(G) might fail to interpolate in certain inequalities. Example 1 Coffman, Hakimi and Schmeichel [4] have proved (as a corollary) the following upper bound on the chromatic number of a connected graph with m edges: The inequality that shows that Hadwiger's number is interpolating in inequality ( 1) is stated explicitly and proved in [5]. Proving that a parameter is interpolating strengthens, of course, the corresponding inequalities for the chromatic number, provided Hadwiger's conjecture is true. Thus, in an intuitive sense, interpolating inequalities are weaker than non-interpolating ones, at least for special graphs, though it is, of course, possible that interpolating inequalities are tight. The purpose of this note is to study the question of interpolation of Hadwiger's number for several upper bounds on the chromatic number due to Ershov and Kozhukin [6], Coffman, Hakimi and Schmeichel [4], Brooks [7], Welsh and Powell [8], and Stacho [9,10]. An useful reference for inequalities concerning the chromatic number is [11]. For Hadwiger's conjecture we refer the reader to [12]. We will use the standard notations K r and C r to denote the complete, respectively cycle graph with r vertices. A graph G is 1-reducible to graph H if the iterative removal of vertices of degree one from G yields graph H. Definition 2 Let G be a graph and e ∈ E(G) be an edge. The contracted graph G e is obtained from G by identifying the endpoints of e and joining the newly created vertex with all vertices that were adjacent to either endpoint. All other vertices and edges are the same as in G. Definition 3 Hadwiger's number h(G) of connected graph G is the largest n such that one can obtain the complete graph K n from G by a series of contractions. Definition 5 For a graph G let V i be the set of vertices of degree i. Also define The following easy result shows that the upper bound on the chromatic number due to Ershov and Kuzhukin [6] is interpolating: Theorem 1 Let G be a connected graph with n vertices and m edges. Then On the other hand, the following result shows that many upper bounds on the chromatic number are non-interpolating: Theorem 2 Hadwiger's number is not interpolating in any of the following inequalities for the chromatic number: [10]), even when restricted to the class of 3-colorable graphs. In [4] the following inequality that improves the Ershov-Kozhukhin bound on the chromatic number was proved: if G is a connected graph that is not 1-reducible to a clique or an odd cycle then χ(G) . For Hadwiger's number the improved upper bound does not quite hold. A class of counterexamples is presented in Figure 1 (a) (with n = 8, m = 10, h(G) = 4). It consists of four trees (edges AB, CD, EF, and GH) with connections between all trees such that no vertex is left with degree one. A similar construction works in the general case. There is a simple technical explanation for the existence of these counterexamples: in the proof of the result from [4] Brooks' inequality is used which, as we have seen, is not interpolating. Nevertheless, we can recover the upper bound, making it interpolating by restricting the class of graphs it applies to: Definition 6 An acyclic contraction of graph G is a contraction that shrinks no cycle of G to a single edge. For example, no edge contraction in a complete graph K n , n ≥ 3 is an acyclic contraction, since it shrinks a triangle to a single edge. On the other hand any contraction in a tree is acyclic. In particular contracting edges AB, CD, EF, and GH in the graph in Figure 1 is an acyclic contraction to complete graph K 4 . We note that any contraction specified by a 1-reducibility is acyclic. Observation 1 If a graph G has an acyclic contraction to an odd cycle C 2t+1 then (by additionally contracting parts of this cycle) one can obtain an acyclic contraction to C 3 ≡ K 3 . Thus we don't really need odd cycles as a final target for acyclic contractions, avoiding acyclic contraction to a complete graph covers this case too. With these preliminaries we can state: Theorem 3 Let G be a connected graph with n vertices and m edges. If G does not have an acyclic contraction to a complete graph K r then Let us consider the question of obtaining interpolating upper bounds on Hadwiger's number in a graph where we know the degree sequence d 1 ≥ d 2 ≥ . . . ≥ d n . Our results suggest an interesting difference between the interpolating and non-interpolating upper bounds: in the upper bounds in Theorem 1 and 3 the upper bounds depends on the degree sequence through quantity m -n = i (d i /2-1). On the other hand the non-interpolating upper bounds are "local": they essentially depend only on one of the degrees (chosen according to the specific condition of each inequality). Of course, one can transform the sequence i (d i /2 -1) to a "local" upper bound n • (∆(G)/2 -1). However, this is somewhat "cheating". The question remains whether one can formalize the notion of "local" upper bounds in a way that excludes these trivial examples, and such that all tight interpolating upper bounds are not local (depend on Ω(n) terms in the sequence d i ). Such a statement is indeed true for our first result, given that interpolating inequalities in Theorem 1 and 3 are tight for all pairs (m, n), n < m < n 2 (see [4]). This easily extends to the following result, which explains why some of the bounds we considered are not interpolating. • Hadwiger's number is interpolating for the above inequality. • Class C contains graphs D r,n from Theorem 2 Then sup {G∈C;d1,...,dn≤3} . It is easy to see that the following are true: 2. For every i = j there exists at least one edge connecting a vertex in V i to a vertex in V j . This happens because the contracted graph K h(G) is a complete graph. We have i.e. Thus h 2 (G) -3h(G) -2(m -n) ≤ 0, and the result immediately follows. For r ≥ 3 we construct a graph D r,n with n nodes, described as follows (and displayed for r = 3, n = 9 in Figure 1(b)): 1. Nodes are labelled 0 to n -1. 2. For all 1 ≤ i, 0 ≤ j ≤ r -1 such that ri + j ≤ n we connect node r • i + j to node r • (i -1) + j. 3. For all 0 ≤ i = j for which ri + j, rj + i < n we connect nodes r • i + j and r • j + i. For all 0 ≤ j ≤ r -1 we can contract all nodes r • i + j into a single node ĵ, by successively contracting edges (of type (ii)) (r • (i -1) + j, r • i + j). The contracted graph has r nodes, and any two of its nodes î = ĵ are connected, because of the edge between nodes r • i + j and r • j + i in graph D r . Thus the contracted graph is the complete graph K r , thus h(D r ) ≥ r. One can verify that, because of the way graph D r,n is constructed d(v) ≤ 3 for every v ∈ V (G). Thus ∆(D r,n ) + 1 = 4, ∆ 2 (D r,n ) + 1 = 4 as well. As for the last bound, it is certainly no larger than ∆(D r,n ) + 2 = 5. Graph D r,n is 3-colorable, for any r ≥ 3, because it is 2-degenerate: in the natural ordering of the vertices each vertex is adjacent to at most two earlier nodes. Thus the GREEDY algorithm 3-colors D r,n . To obtain the result we have to prove the inequality Indeed, in the decomposition of graphs corresponding to obtaining K h(G) at least one of the following situations holds: 1. At least one component corresponding to a vertex is not a tree (including single points among trees), or 2. There exist two components connected by at least two edges. Indeed, suppose neither (1) nor (2) applies. Then by contracting each tree component we would get an acyclic contraction to a K h(G) (including, possibly, the odd cycle C 3 ). Thus m counts at least one extra edge not counted on the right hand side of equation ( 5), so equation ( 6) holds. Now an argument similar to that in the proof of Theorem 1 completes the proof.