Let Σ = (X,B) a 4-cycle system of order v = 1 + 8k. A c-colouring of type s is a map φ∶ B → C, with C set of colours, such that exactly c colours are used and for every vertex x all the blocks containing x are coloured exactly with s colours. Let 4k = qs+r, with q, r ≥ 0. φ is equitable if for every vertex x the set of the 4k blocks containing x is parted in r colour classes of cardinality q + ...

Corrádi and Hajnal proved that for every k ≥ 1 and n ≥ 3k, every graph with minimum degree at least 2k contains k vertex-disjoint cycles. This implies that every 3kvertex graph with maximum degree at most k − 1 has an equitable k-coloring. We prove that for s ∈ {3, 4} if an sk-vertex graph G with maximum degree at most k has no equitable k-coloring, then G either contains Kk+1 or k is odd and G...

The study of graph labeling has focussed on finding classes of graphs which admits a particular type of labeling. In this paper we consider a particular class of graph which admits a vertex magic total labeling. The class we considered here is the class of complete graphs, Kn . A vertex magic labeling of a graph is a bijection which maps the vertices V and edges E to the integers from 1, 2, 3, ...

An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . , m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In this paper we discuss antimagic properties of graphs which contain vert...

A finite directed labeled graph G with vertices {vi)~=l and labels {~)~i~ is solvable with delay d, if for every sequence of labels c h -.ok one can determine a sequence of vertices v% "" v~ such that there is a path connecting the k vertices vt ° ... v~ k labeled g~ ..~ , and for every j v~j is determined by the subsequence a~-+t ... oJ+a only. The paper presents a family of counterexamples pr...

A total labelling of a graph with v vertices and e edges is a one-to-one map taking the vertices and edges onto the set {1, 2, 3, . . . , v + e}. A labelling can be used to define a weight for each vertex and edge. For a vertex the weight is the sum of the label of the vertex and the labels of the incident edges. For an edge {x, y} the weight is the sum of the label of the edge and the labels o...

We consider the problem of learning a labeled graph from a given family of graphs on n vertices in a model where the only allowed operation is to query whether a set of vertices induces an edge. Questions of this type are motivated by problems in molecular biology. In the deterministic nonadaptive setting, we prove nearly matching upper and lower bounds for the minimum possible number of querie...

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is local antimagic if vertex-weights are distinct for pair neighboring vertices, where vertex-weight sum labels all edges incident with that vertex. Any induces proper vertex coloring G color its vertex-weight. This naturally leads to concept chromatic number. The number defined be minimum colors take...