A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that ...

Let G be a graph. A proper vertex coloring of G is said to be a dynamic coloring if for every v ∈ V (G) of degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number of G and is denoted by χ2(G). It was conjectured that if G is an r-regular graph, then χ2(G)−χ(G) ≤ 2. In this tal...

Let G be a multipartite multigraph without loops. Then G is said to be internally fair if its edges are shared as evenly as possible among all pairs of its partite sets. An internally fair factorization of G is an edge-decomposition of G into internally fair regular spanning subgraphs. A holey factor of G is a regular subgraph spanning all vertices but one partite set. An internally fair holey ...

We prove a conjecture of Medvedev and Scanlon [MS14] in the case of regular morphisms of semiabelian varieties. That is, if G is a semiabelian variety defined over an algebraically closed field K of characteristic 0, and φ : G → G is a dominant regular self-map of G which is not necessarily a group homomorphism, we prove that one of the following holds: either there exists a non-constant ration...

It is shown that every (2p+ 1) log2(|V (G)|)-edge-connected graph G has a mod (2p+ 1)orientation, and that a (4p+ 1)-regular graph G has a mod (2p+ 1)-orientation if and only if V (G) has a partition (V , V −) such that ∀U ⊆ V (G), |∂G(U)| ≥ (2p+ 1)||U ∩ V | − |U ∩ V −||. These extend former results by Da Silva and Dahad on nowhere zero 3-flows of 5-regular graphs, and by Lai and Zhang on highl...

For a given a permutation group G, the problem of determining which regular digraphs admit G as an arc-regular group of automorphism is considered. Groups which admit such a representation can be characterized in terms of generating sets satisfying certain properties, and a procedure to manufacture such groups is presented. The technique is based on constructing appropriate factorizations of (s...

We consider commutative regular and context-free grammars, or, in other words, Parikh images of regular and context-free languages. By using linear algebra and a branching analog of the classic Euler theorem, we show that, under an assumption that the terminal alphabet is fixed, the membership problem for regular grammars (given v in binary and a regular commutative grammar G, does G generate v...

According to the classical Friendship Theorem, if G is a finite simple graph such that each two distinct vertices have a unique common neighbor, then G has a dominating vertex. Jian Shen showed that if require only non-neighbors to have a unique common neighbor, then one of the following possibilities hold: i) G has dominating vertex, or ii) G is a strongly regular graph, or iii) G is a bi-regu...

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G v of each vertex Š is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incid...

For every h ∈ N, a graph G with the vertex set V (G) and the edge set E(G) is said to be h-magic if there exists a labeling l : E(G) → Zh\{0} such that the induced vertex labeling s : V (G) → Zh, defined by s(v) = ∑ uv∈E(G) l(uv) is a constant map. When this constant is zero, we say that G admits a zero-sum h-magic labeling. The null set of a graph G, denoted by N(G), is the set of all natural ...