An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families – balanced, balanceable, and unbalanceable – and we obtain a complete classificati...

We obtain several sufficient conditions on the degrees of an oriented graph for the existence of long paths and cycles. As corollaries of our results we deduce that a regular tournament contains an edge-disjoint Hamilton cycle and path, and that a regular bipartite tournament is hamiltonian. An orientedgraph is a directed simple graph, that is to say, a digraph without loops, mutiple arcs, or c...

We describe a set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely coordinatizable yields a simple polygon whose visibility graph is isomorphic to the given graph.

Hugo Hadwiger proved that a graph that is not 3-colorable must have a K4minor and conjectured that a graph that is not k-colorable must have a Kk+1minor. By using the Hochstättler-Nešetřil definition for the chromatic number of an oriented matroid, we formulate a generalized version of Hadwiger’s conjecture that might hold for the class of oriented matroids. In particular, it is possible that e...

Nešeťril and Sopena introduced a concept of oriented game chromatic number and developed a general technique for bounding this parameter. In this paper, we combine their technique with concepts introduced by several authors in a series of papers on game chromatic number to show that for every positive integer k, there exists an integer t so that if C is a topologically closed class of graphs an...

For a vertex v of a connected oriented graph D and an ordered set W = {w1, w2, . . . , wk} of vertices of D, the (directed distance) representation of v with respect to W is the ordered k-tuple r(v ∣ ∣ W ) = (d(v,w1), d(v, w2), . . . , d(v, wk)), where d(v, wi) is the directed distance from v to wi. The set W is a resolving set for D if every two distinct vertices of D have distinct representat...

A graph G is k–weighted–list–antimagic if for any vertex weighting ω : V (G) → R and any list assignment L : E(G)→ 2R with |L(e)| ≥ |E(G)|+k there exists an edge labeling f such that f(e) ∈ L(e) for all e ∈ E(G), labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this pa...

A graph G is said to have a parity-linked orientation φ if every even cycle C2k in G is evenly (resp. oddly) oriented whenever k is even (resp. odd). In this paper, this concept is used to provide an affirmative answer to the following conjecture of D. Cui and Y. Hou [D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, The Electronic J. Combin. 20(2):#P19, 2013]: Let G = G(X,Y ...