We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on $N$ vertices with $\left(\frac{N}{2}\right)-o\left(N^2\right)$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1-o\left(1\right)}$. The second construction provides a covering of all edges of the complete graph $K_N$ by two graphs, each being the edge disjoint union of at most $N^{2-\delta }$ induced matchings, where $\delta >0,076$. This disproves (in a strong form) a conjecture of Meshulam, substantially...

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σv∈V f(v), and the broadcast number λ b (G) is the minimum cost of a dominating broadcast. A set X ⊆ V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The...

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turán [8], we present a sharp lower bound on Kr+1-free graphs for r ≥ 2. Applying the concept of total limited packing we bound the signed total domination number of G with δ(G) ≥ 3 from above by [...] . Also, we prove that γst(T) ≤ n − 2(s − s′) for any tree T of order n, with s support vertices and s′ support vertices...

A maximum matching of a graph $G$ is a matching of $G$ with the largest number of edges. The matching number of a graph $G$, denoted by ${\alpha }^{\text{'}}\left(G\right)$, is the number of edges in a maximum matching of $G$. In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture is true for...

So far, the smallest complete bipartite graph which was known to have a cyclic decomposition into cubes $Q_d$ of a given dimension d was $K_{d{2}^{d-1},d{2}^{d-2}}$. We improve this result and show that also $K_{d{2}^{d-2},d{2}^{d-2}}$ allows a cyclic decomposition into $Q_d$. We also present a cyclic factorization of $K_{8,8}$ into Q₄.