We show that any connected regular graph with d + 1 distinct eigenvalues and odd-girth 2d + 1 is distance-regular, and in particular that it is a generalized odd graph. 2010 Mathematics Subject Classification: 05E30, 05C50; JEL Classification System: C0

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...

Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k−ǫ) poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw(G) admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false. We show here that nevertheless, when k > ⌊∆/2⌋+1, where ∆ is the maximum degree in the graph G, there is a better algorithm, at least when...

A Skolem sequence is a sequence s1, s2, . . . , s2n (where si ∈ A = {1 . . . n}), each si occurs exactly twice in the sequence and the two occurrences are exactly si positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The problem of deciding which sets of the form A = {1 . . . n} are Skolem sets was solved by Thoralf Skolem in the late 1950’s. We stud...

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 ...