Lattices of relative colour-families and antivarieties

European Journal of Combinatorics, 1985

A class J{ of relational systems (of the same type) has the 8-Ramsey property if for every $ E J{ there is T E J{ such that to every 2-coloring of (!) (= relational subsystems of T isomorphic to 8) we can find a monochromatic (iJ for some {J E (D. Extending recent results by JeZek and Ndetnl we prove it for (a) every class J{ of finite reflexive relational systems closed for products and 8 E J{ a singleton, (b) every abstract class J{ of finite relational systems with the strong amalgamation property and 8 E J{ such that the sets from (!) are disjoint for all $ E J{. Finally we prove: Let J{ be an abstract class of finite reflexive or areflexive relational systems with the strong amalgamation property. If J{ has the 8-Ramsey property, then 8 is constant.

Cornell University - arXiv, 2022

In this paper, we characterize chordal and perfect zero-divisor graphs of finite posets. Also, it is proved that the zero-divisor graphs of finite posets and the complement of zero-divisor graphs of finite 0-distributive posets satisfy the Total Coloring Conjecture. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graph of rings, the annihilating ideal graphs, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of cyclic groups. In fact, it is proved that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

Mathematica Bohemica, 2002

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Journal of Relation …, 2004

Journal of Combinatorial Theory, 1999

In this paper, we introduce a measure of the extent to which a finite combinatorial structure is a Ramsey object in the class of objects with a similar structure. We show for classes of finite relational structures, including graphs, binary posets, and bipartite graphs, how this measure depends on the symmetries of the structure.

Mathematica Slovaca, 2010

The concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if such a graph is derived from an atomic or distributive lattice, then the chromatic number equals the clique number. If this number is finite, then in the case of a distributive lattice, it is determined by the number of minimal prime ideals in the lattice. An estimate for the number of edges in such a graph of a finite lattice is given.

European Journal of Combinatorics, 2014

A relational structure is homomorphism-homogeneous (HH-homogeneous for short) if every homomorphism between finite induced substructures of the structure can be extended to a homomorphism over the whole domain of the structure. Similarly, a structure is monomorphism-homogeneous (M H-homogeneous for short) if every monomorphism between finite induced substructures of the structure can be extended to a homomorphism over the whole domain of the structure. In this paper we consider L-colored graphs, that is, undirected graphs without loops where sets of colors selected from L are assigned to vertices and edges. A full classification of finite M H-homogeneous L-colored graphs where L is a chain is provided, and we show that the classes M H and HH coincide. When L is a diamond, that is, a set of pairwise incomparable elements enriched with a greatest and a least element, the situation turns out to be much more involved. We show that in the general case the classes M H and HH do not coincide.

Discrete Mathematics, 1996

The aim of this paper is to study those pairs of complementary equivalence relations on a fixed set which are maximal as families of mutually complementary equivalence relations. The existence of such pairs on uncountable sets was proved by Steprgns and Watson (1995). They conjectured that such pairs do not exist in the finite and in the countable case. Here we disprove this conjecture by proving that they exist in huge quantity in both cases. We study in detail the case when: (a) one of the equivalence relations in the pair has precisely two equivalence classes; (b) one of the equivalence relations has at most three equivalence classes; (c) in one of the equivalence relations all but one equivalence classes are singletons. In cases (a) and (c) we describe all pairs of complementary equivalence relations having this extremal property. In the case (a) the non-extremal pairs are related to Turan graphs.

Discrete Applied Mathematics, 2021

Indicated coloring is a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by χ i (G). In this paper, we have shown that for any graphs G and (H). Also, we have shown that for any graph G and for some classes of graphs H with χ(H) As a consequence of this result we have shown that if G ∈ G = Chordal graphs, Cographs, Complement of bipartite graphs, {P 5 , C 4 }-free graphs, connected {P 6 , C 5 , P 5 , K 1,3 }-free graphs which contain an induced C 6 , Complete multipartite graphs and H ∈ F = Bipartite graphs, Chordal graphs, Cographs, {P 5 , K 3 }-free graphs, {P 5 , P aw}-free graphs, Complement of bipartite graphs, {P 5 , K 4 , Kite, Bull}-free graphs, connected {P 6 , C 5 , P 5 , K 1,3 }-free graphs which contain an induced C 6 , K[C 5 ](m 1 , m 2 , . . . , m 5 ), {P 5 , C 4 }-free graphs, connected {P 5 , P 2 ∪ P 3 , P 5 , Dart}free graphs which contain an induced ). This serves as a partial answer to one of the questions raised by A. Grzesik in . In addition, if G is a Bipartite graph or a {P 5 , K 3 }-free graph (or) a {P 5 , P aw}-free graph and H ∈ F, then we have shown that χ i (G[H]) = χ(G[H]).

Algebra universalis, 2011

We introduce "π-versions" of five familiar conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D 0π if a∧ (b ∨ c) (a∧ b)∨ c for all 3-element antichains {a, b, c}. We consider a congruence relation ∼ whose blocks are the maximal autonomous chains and define the order-skeleton of a lattice L to be L := L/∼. We prove that the following are equivalent for a lattice L: (i) L satisfies D 0π , (ii) L satisfies any of the five π-versions of distributivity, (iii) the order-skeleton L is distributive.