Cores and Compactness of Infinite Directed Graphs

AbstractIn this paper we define the property of homomorphic compactness for digraphs. We prove that if a digraphHis homomorphically compact thenHhas a core, although the converse does not hold. We also examine a weakened compactness condition and show that when this condition is assumed, compactness is equivalent to containing a core. We use this result to prove that if a digraphHof sizeκis not compact, then there is a digraphGof size at mostκ+such thatHis not compact with respect toG. We then give examples of some sufficient conditions for compactness

Homomorphic Compactness of Infinite Graphs

The University of CalgaryIn 1951 de Bruijn and Erdös proved that an infinite graph is n-colourable if and only if each of its finite subgraphs is n- colourable. This is often referred to as \u27compactness of n-colouring\u27. Using the fact that n-colouring is essentially identical to finding a graph homomorphism to a complete graph on n vertices, we say that a graph G is homomorphically compact if each infinite graph H admits a homomorphism to G exactly when all of its finite subgraphs admit such a homomorphism. We will show that (really) infinite compact graphs exist and explore various other problems related to them

Generating Alternating Permutations Lexicographically

Abstract A permutation ss1ss2 \Delta \Delta \Delta ssn is alternating if ss1! ss2? ss3! ss4 \Delta \Delta \Delta. We present a constant average-time algorithm for generating all alternating permutations in lexicographic order. Ranking and unranking algorithms are also derived