We explore the connection between locally constrained graph homomorphisms and degree matrices arising from an equitable partition of a graph. We provide several equivalent characterizations of degree matrices. As a consequence we can efficiently check whether a given matrix M is a degree matrix of some graph and also compute the size of a smallest graph for which it is a degree matrix in polyno...

For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (G), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value ω(f) = ∑ v∈V f(v). The total {k}-dominati...

Let D = (V,A) be a finite simple directed graph (shortly digraph) in which dD(v) ≥ 1 for all v ∈ V . A function f : V −→ {−1, 1} is called a signed total dominating function if ∑ u∈N−(v) f(u) ≥ 1 for each vertex v ∈ V . A set {f1, f2, . . . , fd} of signed total dominating functions on D with the property that ∑d i=1 fi(v) ≤ 1 for each v ∈ V (D), is called a signed total dominating family (of f...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a vertex with label 2 within distance k from each other. A set {f1, f2, . . . , fd} of k-distance Roman dominating functions on G with the property that ∑d i=1 fi(v) ≤ 2 for each v ∈ V (G), is call...

We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a structure has an equitable regularity partition of size polynomial in the reciprocal of the desired accuracy, and such that for each k-ary relation and k-tuple of ...

A C4-decomposition of Kv−F , where F is a 1-factor of Kv and C4 is the cycle of length 4, is a partition P of E(Kv − F ) into sets, each element of which induces a C4 (called a block). A function assigning a color to each block defined by P is said to be an (s, p)-equitable block-coloring if: exactly s colors are used; each vertex v is incident with blocks colored with exactly p colors; and the...

We prove that a minimal automaton has a minimal adjacency matrix rank and a minimal adjacency matrix nullity using equitable partition (from graph spectra theory) and Nerode partition (from automata theory). This result naturally introduces the notion of matrix rank into a regular language L, the minimal adjacency matrix rank of a deterministic automaton that recognises L. We then define and fo...

Using a suitable orientation, we give a short proof of a result of Czumaj and Strothmann [3]: Every 2-edge-connected graph G contains a spanning tree T with the property that dT (v) ≤ dG(v)+3 2 for every vertex v. Trying to find an analogue of this result in the directed case, we prove that every 2-arc-strong digraph D has an out-branching B such that d+B(x) ≤ d+D(x) 2 + 1. As a corollary, ever...

In the fractional graph colouring problem, the task is to schedule the activities of the nodes so that each node is active for 1 time unit in total, and at each point of time the set of active nodes forms an independent set. We show that for any α > 1 there exists a deterministic distributed algorithm that finds a fractional graph colouring of length at most α(∆ + 1) in any graph in one synchro...