The natural conjecture, formulated in several sources, asserts that the H-coloring problem is NP-complete for any non-bipartite graph H, and a proof of this conjecture is given.Expand

This chapter discusses the structure of Composition, the partial order of Graphs and Homomorphisms, and testing for the Existence of Homomorphism.Expand

This is the first book devoted to the systematic study of sparse graphs and sparse finite structures and the authors devised an unifying classification of general classes of structures that is very robust and has many remarkable properties.Expand

The notions tree-depth and upper chromatic number of a graph are defined and it is shown that the upper chromatics number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs.Expand

For classes of graphs with bounded expansion with a new invariant, the greatest reduced average density of G with rankr, @?"r(G) is introduced, it is proved the existence of several partition results such as theexistence of low tree-width and lowTree-depth colorings.Expand

The main results are partial characterizations of countable graphs and posets with this property; an analogue of Fraïssé's theorem; and representations of monoids as endomorphism monoids of such structures.Expand

The nowhere dense versus somewhere dense dichotomy is expressed in terms of edge densities as a trichotomy theorem and has a number of applications to mathematical logic, complexity of algorithms, and combinatorics.Expand

It is found that the core of a finite graph is unique (up to isomorphism) and is also its smallest retract, and some homomorphism properties of cores are investigated, concluding that it is NP-complete to decide whether a graph is its own core.Expand

Abstract The following theorem is proved: Let G be a finite graph with cl( G ) = m , where cl( G ) is the maximum size of a clique in G . Then for any integer r ≥ 1, there is a finite graph H , also… Expand