Combinatorica最新文献

The defective chromatic number of a graph class is the infimum k such that there exists an integer d such that every graph in this class can be partitioned into at most k induced subgraphs with maximum degree at most d. Finding the defective chromatic number is a fundamental graph partitioning problem and received attention recently partially due to Hadwiger’s conjecture about coloring minor-closed families. In this paper, we prove that the defective chromatic number of any minor-closed family equals the simple lower bound obtained by the standard construction, confirming a conjecture of Ossona de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable finite minors for infinite graphs that cannot be partitioned into a fixed finite number of induced subgraphs with uniformly bounded maximum degree. As corollaries about clustered coloring, we obtain a linear relation between the clustered chromatic number of any minor-closed family and the tree-depth of its forbidden minors, improving an earlier exponential bound proved by Norin, Scott, Seymour, and Wood and confirming the planar case of their conjecture.

In this paper we prove that every totally real algebraic integer (lambda ) of degree (d ge 2) occurs as an eigenvalue of some tree of height at most (d(d+1)/2+3). In order to prove this, for a given algebraic number (alpha ne 0), we investigate an additive semigroup that contains zero and is closed under the map (x mapsto alpha /(1-x)) for (x ne 1). The problem of finding the smallest such semigroup seems to be of independent interest.

Abstract

Given a (thick) irreducible spherical building (Omega ) , we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of (Omega ) (in particular the residue of a vertex corresponding to a point of the long root geometry, but also other types of vertices occur). We apply this to the finite case to obtain new results on the generating rank of mainly the exceptional long root geometries, answering an open question by Cooperstein about the generating ranks of the exceptional long root subgroup geometries. We completely settle the finite case for long root geometries of type ({{textsf{A}}}_n) , and the case of type (mathsf {F_{4,4}}) over any field with characteristic distinct from 2 (which is not a long root subgroup geometry, but a hexagonic geometry).

For strongly connected, pure n-dimensional regular CW-complexes, we show that evenness (each ((n{-}1))-cell is contained in an even number of n-cells) is equivalent to generalizations of both cycle decomposition and traversability.

Abstract

We show that every graph with pathwidth strictly less than a that contains no path on (2^b) vertices as a subgraph has treedepth at most 10ab. The bound is best possible up to a constant factor.

For (0 le t le r) let m(t, r) be the maximum number s such that every t-edge-connected r-graph has s pairwise disjoint perfect matchings. There are only a few values of m(t, r) known, for instance (m(3,3)=m(4,r)=1), and (m(t,r) le r-2) for all (t not = 5), and (m(t,r) le r-3) if r is even. We prove that (m(2l,r) le 3l - 6) for every (l ge 3) and (r ge 2 l).

A topological version of the famous Hedetniemi conjecture says: The mapping index of the Cartesian product of two ({mathbb {Z}}/2)- spaces is equal to the minimum of their ({mathbb {Z}}/2)-indexes. The main purpose of this article is to study the topological version of the Hedetniemi conjecture for G-spaces. Indeed, we show that the topological Hedetniemi conjecture cannot be valid for general pairs of G-spaces. More precisely, we show that this conjecture can possibly survive if the group G is either a cyclic p-group or a generalized quaternion group whose size is a power of 2.

Fuglede’s conjecture states that a subset (Omega subseteq mathbb {R}^{n}) with positive and finite Lebesgue measure is a spectral set if and only if it tiles (mathbb {R}^{n}) by translation. However, this conjecture does not hold in both directions for (mathbb {R}^n), (nge 3). While the conjecture remains unsolved in (mathbb {R}) and (mathbb {R}^2), cyclic groups are instrumental in its study within (mathbb {R}). This paper introduces a new tool to study spectral sets in cyclic groups and, in particular, proves that Fuglede’s conjecture holds in (mathbb {Z}_{p^{n}qr}).

Let I(G) be the edge ideal of a simple graph G. We prove that (I(G)^2) has a linear free resolution if and only if G is gap-free and ({{,textrm{reg},}}I(G) le 3). Similarly, we show that (I(G)^3) has a linear free resolution if and only if G is gap-free and ({{,textrm{reg},}}I(G) le 4). We deduce these characterizations by establishing a general formula for the regularity of powers of edge ideals of gap-free graphs ({{,textrm{reg},}}(I(G)^s) = max ({{,textrm{reg},}}I(G) + s-1,2s)), for (s =2,3).

Huynh et al. recently showed that a countable graph G which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds if G contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and Kühn. Moreover, we construct a locally finite planar graph which contains every locally finite planar graph as a topological minor. This shows that in the above result it is not enough to require that G contains every locally finite planar graph as a topological minor.