On -bounded families of graphs.
The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K₁ by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ-graphs.
Let P denote a 3-uniform hypergraph consisting of 7 vertices a, b, c, d, e, f, g and 3 edges {a, b, c}, {c, d, e}, and {e, f, g}. It is known that the r-color Ramsey number for P is R(P; r) = r + 6 for r ≤ 9. The proof of this result relies on a careful analysis of the Turán numbers for P. In this paper, we refine this analysis further and compute the fifth order Turán number for P, for all n. Using this number for n = 16, we confirm the formula R(P; 10) = 16.
Adhesive polymatroids were defined by F. Matúš motivated by entropy functions. Two polymatroids are adhesive if they can be glued together along their joint part in a modular way; and are one-adhesive, if one of them has a single point outside their intersection. It is shown that two polymatroids are one-adhesive if and only if two closely related polymatroids have joint extension. Using this result, adhesive polymatroid pairs on a five-element set are characterized.
We investigate congruences in one-element extensions of algebras in the variety generated by tournaments.
For a given induced hereditary property 𝓟, a 𝓟-coloring of a graph G is an assignment of one color to each vertex such that the subgraphs induced by each of the color classes have property 𝓟. We consider the effectiveness of on-line 𝓟-coloring algorithms and give the generalizations and extensions of selected results known for on-line proper coloring algorithms. We prove a linear lower bound for the performance guarantee function of any stingy on-line 𝓟-coloring algorithm. In the class of generalized...
The unit square can be on-line covered with any sequence of squares whose total area is not smaller than 4.
In on-line computation, the instance of the problem dealt is not entirely known from the beginning of the solution process, but it is revealed step-by-step. In this paper we deal with on-line independent set. On-line models studied until now for this problem suppose that the input graph is initially empty and revealed either vertex-by-vertex, or cluster-by-cluster. Here we present a new on-line model quite different to the ones already studied. It assumes that a superset of the final graph is initially...
If n ≥ 3, then any sequence of squares of side lengths not greater than 1 whose total area does not exceed ¼(n+1) can be on-line packed into n unit squares.
A k-ranking of a graph G is a colouring φ:V(G) → 1,...,k such that any path in G with endvertices x,y fulfilling φ(x) = φ(y) contains an internal vertex z with φ(z) > φ(x). On-line ranking number of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an arbitrary order; when colouring a vertex, only edges between already present vertices are known. Schiermeyer, Tuza and Voigt proved that for n ≥ 2. Here we show...