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We characterise gaps in the full homomorphism order of graphs., Comment: 9 pages, extended abstract for Eurocomb 2017

We show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this "fractal" property contributes to the spectacular properties of the homomorphism order. We first show the fractal property by using Sparse Incomparability Lemma and then by more involved elementary argument., Comment: 8 pages, 5 figures. Minor corrections and reformatting. Accepted to European Journal of Combinatorics

We prove that for every $d\geq 3$ the homomorphism order of the class of line graphs of finite graphs with maximal degree $d$ is universal. This means that every finite or countably infinite partially ordered set may be represented by line graphs of graphs with maximal degree $d$ ordered by the existence of a homomorphism., Comment: 13 pages, 8 figures, accepted to European Journal of Combinatorics

Given two graphs G and H, we ask under which conditions there is a relation R that generates the edges of H given the structure of graph G. This construction can be seen as a form of multihomomorphism. It generalizes surjective homomorphisms of graphs and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time., Comment: accepted by Ars Mathematica Contemporanea