Let G be a finite graph with p vertices and x its chromatic polynomial. A combinatorial interpretation is given to the positive integer (-l)px(-A), where h is a positive integer, in terms of acyclic orientations of G. In particular, (-l)Px(-1) is the number of acyclic orientations of G. An application is given to the enumeration of labeled acyclic digraphs. An algebra of full binomial type, in ...

This paper arose from attempting to understand Bousfield localization functors in stable homotopy theory. All spectra will be p-local for a prime p throughout this paper. Recall that if E is a spectrum, a spectrum X is Eacyclic if E∧X is null. A spectrum is E-local if every map from an E-acyclic spectrum to it is null. A map X → Y is an E-equivalence if it induces an isomorphism on E∗, or equiv...

The Classification Problem is the problem of deciding whether a simple graph has chromatic index equals to ∆ or ∆+1, where ∆ is the maximum degree of the graph. It is known that to decide if a graph has chromatic index equals to ∆ is NP-complete. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The chromatic indexes for some subsets of split graphs, s...

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arbp(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|, p + 1) colours. In the particular c...

A graph is dually chordal if it is the clique graph of a chordal graph. Alternatively, a graph is dually chordal if it admits a maximum neighbourhood order. This class generalizes known subclasses of chordal graphs such as doubly chordal graphs, strongly chordal graphs and interval graphs. We prove that Vizing's total-colour conjecture holds for dually chordal graphs. We describe a new heuristi...

Abstract We give the first examples of $${\mathcal {O}}$$ O -acyclic smooth projective geometrically connected varieties over function field a complex curve, whose index is not equal to one. More precisely, we construct family Enriques surfaces $${\mathbb {P}}^{1}$$ P </...

The b-chromatic index φ(G) of a graph G is the largest integer k such that G admits a proper k-edge coloring in which every color class contains at least one edge incident to some edge in all the other color classes. The b-chromatic index of trees is determined and equals either to a natural upper bound m(T ) or one less, where m(T ) is connected with the number of edges of high degree. Some co...

We study edge coloring games defining the so-called game chromatic index of a graph. It has been reported that the game chromatic index of trees with maximum degree ∆ = 3 is at most ∆ + 1. We show that the same holds true in case ∆ ≥ 6, which would leave only the cases ∆ = 4 and ∆ = 5 open.

The automorphic A-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by a given subgroup A of the full automorphism group of Γ. We compute the automorphic A-chromatic index of each generalized Petersen graph when A is the full automorphism group.

We study quasipolynomials enumerating proper colorings, nowherezero tensions, and nowhere-zero flows in an arbitrary CW-complex X, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in Z/kZ for some k) or integral (with values in {−k + 1, . . . , k − 1}). We obtain deletion-contraction recurrences and closed ...