A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has ...

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The c...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial k-trees, i.e., graphs with bounded tree-width. Howeve...

BACKGROUND Model-based virtual screening plays an important role in the early drug discovery stage. The outcomes of high-throughput screenings are a valuable source for machine learning algorithms to infer such models. Besides a strong performance, the interpretability of a machine learning model is a desired property to guide the optimization of a compound in later drug discovery stages. Linea...

A proper vertex coloring of a graph is called linear if the subgraph induced by the vertices colored by any two colors is a set of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by lc(G), is the minimum number of colors in a linear coloring of G. Extending a result of Alon, McDiarmid and Reed concerning acyclic graph colorings, we show that if G has maximum degree d th...

We give a linear-time algorithm to decide 3-colorability of a trianglefree graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the coloring for a bounded number of vertices.

Among all collections of a given number of k-element subsets of an n-element groundset find a collection which maximizes the number of pairs of subsets which intersect in k − 1 elements. This problem was solved for k = 2 by Ahlswede and Katona, and is open for k > 2. We survey some linear algebra approaches which yield to estimations for the maximum number of pairs, and we present a new and sho...

Let V be 2n-dimensional vector space over a field K equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let K0 ⊆ K be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, . . . , 2n}, there exist natural representations of the groups G ∼= U(2n,K/K0) and H = G∩SL(V ) ∼= SU(2n,K/K0) on the k-th exterior power of V . With the aid of linear...