Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h : C → G(φ), where G(φ) is a graph associated with φ and C is a coloured...

Bounds on the range of random graph homomorphism into Z, and the maximal height diierence of the Gaussian random eld, are presented.

An oriented graph is a digraph with no opposite arcs. The oriented chromatic number of an oriented graph G⃗ is the minimum order of an oriented graph to which G⃗ admits a homomorphism. The oriented chromatic number of an undirected graph is then defined as the maximum oriented chromatic number of its orientations. In this paper, we survey the main results about this graph parameter and propose a ...

For a fixed graph $H$, by Hom($H$) we denote the computational problem which asks whether given $G$ admits homomorphism to i.e., an edge-preserving mapping from $V(G)$ $V(H)$. As Hom($K_k$) is equivalent $k$-Coloring, homomorphisms can be seen as generalizations of colorings. It known that polynomial-time solvable if $H$ bipartite or has vertex with loop, and NP-complete otherwise [Hell Nešetři...

For digraphs D and H , a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOMP(H). Digraphs are allowed to have loops, but not allowed to have parallel arcs. A natural optimization version of the homomorphism probl...

We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to “subsurface Torelli groups”. Using this, we extend Johnson’s calculation of the rational abelianization of the Torelli group not on...

Twisted homomorphisms of bialgebras are bialgebra homomorphisms from the first into Drinfeld twistings of the second. They possess a composition operation extending composition of bialgebra homomorphisms. Gauge transformations of twists, compatible with adjacent homomorphisms, give rise to gauge transformation of twisted homomorphisms, which behave nicely with respect to compositions. Here we s...

A graph K is called multiplicative if whenever a categorical product of two graphs admits a homomorphism to K, then one of the factors also admits a homomorphism to K. We prove that all circular graphs Kk/d such that k/d < 4 are multiplicative. This is done using semi-lattice endomorphism in (the skeleton of) the category of graphs to prove the multiplicativity of some graphs using the known mu...

Oriented graphs are directed graphs without opposite arcs. In other words an oriented graph is an orientation of an undirected graph, obtained by assigning to every edge one of the two possible orientations. IfG is a graph, V (G) denotes its vertex set, E(G) denotes its set of edges (or arcs if G is an oriented graph) and F (G) denotes its set of faces if G is planar. A homomorphism from an ori...

This paper is based on a course delivered by the author at NCTS, National Chiao Tung University, Taiwan in Febuary 1999. We survey results related to structural aspects of graph homomorphism. Our aim is to demonstrate that this forms today a compact collection of results and methods which perhaps deserve its name : structural combinatorics. Due to space limitations we concentrate on a sample of...