Given a family ℱ of multigraphs without isolated vertices, a multigraph M is called ℱ-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of ℱ. We present necessary and sufficient conditions for existence of such decompositions if ℱ consists of all multigraphs of size q except for one. Namely, for a multigraph H of size q we find each multigraph M of size kq, such that every partition of the edge set of M into parts of cardinality q contains a part...

Given a family 𝓕 of multigraphs without isolated vertices, a multigraph M is called 𝓕-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of 𝓕. We present necessary and sufficient conditions for the existence of such decompositions if 𝓕 comprises two multigraphs from the set consisting of a 2-cycle, a 2-matching and a path with two edges.

An arc decomposition of the complete digraph Kₙ into t isomorphic subdigraphs is generalized to the case where the numerical divisibility condition is not satisfied. Two sets of nearly tth parts are constructively proved to be nonempty. These are the floor tth class ( Kₙ-R)/t and the ceiling tth class ( Kₙ+S)/t, where R and S comprise (possibly copies of) arcs whose number is the smallest possible. The existence of cyclically 1-generated decompositions of Kₙ into cycles ${}^{\to }{C}_{n-1}$ and into paths ${}^{\to }Pₙ$ is characterized....

A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of order n ≥ 22 and with [...] ‖G‖ > (n−42)+12 $\left|\right|G\left|\right|>\left(n-42\right)+12$ edges is AP or belongs to few classes of exceptional graphs.

Let $\sum$ be a flat surface of genus $g$ with cone type singularities. Given a bipartite graph $\Gamma$ isoradially embedded in $\sum$, we define discrete analogs of the $2^{2g}$ Dirac operators on $\sum$. These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair $\Gamma \subset \sum$ for these discrete Dirac operators to be Kasteleyn matrices of the graph $\Gamma$. As a consequence, if these conditions are met, the partition function of the dimer...

We prove that if G is a graph of order 5k and the minimum degree of G is at least 3k then G contains k disjoint cycles of length 5.

Let n, s and t be three integers with s ≥ 1, t ≥ 0 and n = 3s + 4t. Let G be a graph of order n such that the minimum degree of G is at least (n + s)/2. Then G contains a 2-factor with s + t components such that s of them are triangles and t of them are quadrilaterals.

For a set D of positive integers, we define a vertex set S ⊆ V(G) to be D-independent if u, v ∈ S implies the distance d(u,v) ∉ D. The D-independence number ${\beta }_D\left(G\right)$ is the maximum cardinality of a D-independent set. In particular, the independence number $\beta \left(G\right)={\beta }_1\left(G\right)$. Along with general results we consider, in particular, the odd-independence number ${\beta }_{ODD}\left(G\right)$ where ODD = 1,3,5,....

Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper. Among other results, we characterize bipartite and...

Let V₁, V₂ be a partition of the vertex set in a graph G, and let ${\gamma }_i$ denote the least number of vertices needed in G to dominate $V_i$. We prove that γ₁+γ₂ ≤ [4/5]|V(G)| for any graph without isolated vertices or edges, and that equality occurs precisely if G consists of disjoint 5-paths and edges between their centers. We also give upper and lower bounds on γ₁+γ₂ for graphs with minimum valency δ, and conjecture that γ₁+γ₂ ≤ [4/(δ+3)]|V(G)| for δ ≤ 5. As δ gets large, however, the largest possible value...

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number $s{d}_{\gamma }\left(G\right)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that $1\le s{d}_{\gamma }\left(G\right)\le 3$ for any graph G. We give a counterexample to this conjecture. On the other hand, we show...