We study Hamiltonicity and pancyclicity in the graph obtained as union of a deterministic n $$ -vertex H with δ ( ) ≥ α \delta (H)\ge \alpha random d -regular G , for ∈ { 1 2 } d\in \left\{1,2\right\} . When is 2-regular graph, we prove that a.a.s. ∪ H\cup pancyclic all 0 ] \in \left(0,1\right] also extend our result to range sublinear degrees. 1-regular − \left(\sqrt{2}-1,1\right] this best po...

We show that no more new distance-regular graphs in the tables of book (Brouwer, Cohen, Neumaier, 1989) can be produced by using coset graph additive completely regular codes over finite fields.

A model for cleaning a graph with brushes was recently introduced. We consider the minimum number of brushes needed to clean d-regular graphs in this model, focusing on the asymptotic number for random d-regular graphs. We use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method to find the (asymptotic) n...

A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A (κ, τ)-regular set is a subset of the vertices inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbours in it. We consider the case when κ = τ...

A model for cleaning a graph with brushes was recently introduced. We consider the minimum number of brushes needed to clean d-regular graphs in this model, focusing on the asymptotic number for random d-regular graphs. We use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method to find the (asymptotic) n...