Note on parity factors of regular graphs

Elementary Graphs with Respect to f-Parity Factors

This note concerns the f -parity subgraph problem, i.e., we are given an undirected graph G and a positive integer value function f : V (G) → N, and our goal is to find a spanning subgraph F of G with degF ≤ f and minimizing the number of vertices x with degF (x) ≡ f(x) mod 2. First we prove a Gallai–Edmonds type structure theorem and some other known results on the f -parity subgraph problem, ...

D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs

The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated lin...

Note on winning positions on pushdown games with omega-regular winning conditions

We consider infinite two-player games on pushdown graphs. For parity winning conditions, we show that the set of winning positions of each player is regular and we give an effective construction of an alternating automaton recognizing it. This provides a DEXPTIME procedure to decide whether a position is winning for a given player. Finally, using the same methods, we show, for any ω-regular win...

Parity-regular Steinhaus graphs

Steinhaus graphs on n vertices are certain simple graphs in bijective correspondence with binary {0,1}-sequences of length n−1. A conjecture of Dymacek in 1979 states that the only nontrivial regular Steinhaus graphs are those corresponding to the periodic binary sequences 110...110 of any length n − 1 = 3m. By an exhaustive search the conjecture was known to hold up to 25 vertices. We report h...

A Note on Strongly Quotient Graphs