Let G be a finite simple graph with edge ideal I (G). Let I (G)∨ denote the Alexander dual of I (G). We show that a description of all induced cycles of odd length in G is encoded in the associated primes of (I (G)∨)2. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simp...

A graph G is strict quasi parity (SQP) if every induced subgraph of G that is not a clique contains a pair of vertices with no odd chordless path between them (an even pair). Hougardy conjectured that the minimal forbidden subgraphs for the class of SQP graphs are the odd chordless cycles, the complements of odd or even chordless cycles, and some line-graphs of bipartite graphs. Here we prove t...

Two vertices in a graph are called an even pair (odd pair) if all induced paths between these two vertices have even (odd) length. Even and odd pairs have turned out to be of importance in conjunction with perfect graphs. We will characterize all linegraphs of bipartite graphs that contain an even resp. odd pair. In general, it is a co-NP-complete problem to decide whether a graph contains an e...

By a regular embedding of a graph into an orientable surface we mean a 2cell embedding with the automorphism group acting regularly on arcs. In 1997 Nedela and Škoviera [Europ. J. Comb. 18, 807-823] presented a construction giving for each solution of the congruence e2 ≡ 1(mod n) a regular embedding Me of the hypercube Qn. It was conjectured that all regular embeddings of Qn can be constructed ...

We investigate the following very simple load-balancing algorithm on the N-cycle (N even) which we call Odd-Even Transposition Balancing (OETB). The edges of the cycle are partitioned into two matchings canonically. A matching defines a single step, the two matchings form a single round. Processors connected by an edge of the matching perfectly balance their loads, and, if there is an excess to...

Edge (5, 3)-colorings of a (5, 3)-regular bipartite graph relating 5cycles and vertices of the Petersen graph lead to solutions of the Great Circle Challenge. This takes, for odd n = 2k + 1 > 1, to a canonical 1-1 correspondence from the family of n-cycles of the complete graph Kn onto the family Ck of n-cycles of the odd graph Ok+1 and onto the family of their pullback 2n-cycles in the middle-...