Antivarieties and colour-families of graphs

Abstract. We suggest an algebraic approach to the study of colour-families of graphs. This approach is based on the notion of a congruence of an arbitrary structure. We prove that every colour-family of graphs is a finitely generated universal Horn class and show that for every colour-family the universal theory is decidable. We study the structure of the lattice of colour-families of graphs and the lattice of antivarieties of graphs. We also consider bases of quasi-identities and bases of anti-identities for colour-families and find certain relations between the existence of bases of a special form and problems in graph theory.

[1]  Jaroslav Nesetril,et al.  On classes of relations and graphs determined by subobjects and factorobjects , 1978, Discret. Math..

[2]  Frank Harary,et al.  Graph Theory , 2016 .

[3]  S. Burris,et al.  Sheaf constructions and their elementary properties , 1979 .

[4]  Stanley Burris.  Model Companions for Finitely Generated Universal Horn Classes , 1984, J. Symb. Log..

[5]  S. Hedetniemi.  Homomorphisms of graphs and automata , 1967 .

[6]  Norbert Sauer,et al.  The chromatic number of the product of two 4-chromatic graphs is 4 , 1985, Comb..

[7]  A. Pultr,et al.  Productive classes and subdirect irreducibility, in particular for graphs , 1977, Discret. Math..

[8]  Norbert Sauer,et al.  An approach to hedetniemi's conjecture , 1992, J. Graph Theory.

[9]  Derick Wood,et al.  Colorings and interpretations: a connection between graphs and grammar forms , 1981, Discret. Appl. Math..

[10]  Ales Pultr.  On Sabidussi-Fawcett subdirect representation , 1992, Discret. Math..

[11]  V. Gorbunov,et al.  Algebraic theory of quasivarieties , 1998 .

[12]  A. Budkin,et al.  Quasivarieties of algebraic systems , 1975 .

[13]  Xuding Zhu,et al.  Uniquely H -colorable graphs with large girth , 1996 .

[14]  X. Caicedo.  Finitely axiomatizable quasivarieties of graphs , 1995 .

[15]  Arto Salomaa,et al.  On color-families of graphs , 1981 .

[16]  Bruce L. Bauslaugh.  Core-like properties of infinite graphs and structures , 1995, Discret. Math..

[17]  Emo Welzl.  Color-Families are Dense , 1982, Theor. Comput. Sci..

[18]  The first order theory of $N$-colorable graphs , 1979 .

[19]  Pavol Hell,et al.  On multiplicative graphs and the product conjecture , 1988, Comb..

[20]  V. Gorbunov,et al.  Covers in lattices of quasivarieties and independent axiomatizability , 1977 .

[21]  R. McKenzie.  Equational bases and nonmodular lattice varieties , 1972 .

[22]  P. Erdös,et al.  Graph Theory and Probability , 1959 .

[23]  V. Gorbunov,et al.  Universal Horn classes and antivarieties of algebraic systems , 2000 .