For an assignment of numbers to the vertices of a graph, let S[u] be the sum of the labels of all the vertices in the closed neighborhood of u, for a vertex u. Such an assignment is called closed distinguishing if S[u] 6= S[v] for any two adjacent vertices u and v unless the closed neighborhoods of u and v coincide. In this note we investigate dis[G], the smallest integer k such that there is a...

We prove that the strong chromatic index for each k-degenerate graph with maximum degree ∆ is at most (4k − 2)∆ − k(2k − 1) + 1. A strong edge-coloring of a graph G is an edge-coloring so that no edge can be adjacent to two edges with the same color. So in a strong edge-coloring, every color class gives an induced matching. The strong chromatic index χs(G) is the minimum number of colors needed...

A strong edge-coloring of a graph is a function that assigns to each edge a color such that every two distinct edges that are adjacent or adjacent to a same edge receive different colors. The strong chromatic index χs(G) of a graph G is the minimum number of colors used in a strong edge-coloring of G. From a primal-dual point of view, there are three natural lower bounds of χs(G), that is σ(G) ...

It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if G is a connected graph distinct from C7, then there is a χ(G)-coloring of G in which every vertex v ∈ V (G) is an initial vertex of a path P with χ(G) vertices whose colors are different. In [S. Akbari, V. Liaghat, and A. Nikzad. Colorful paths in vertex coloring of...

The adaptable chromatic number of a graph G is the smallest integer k such that for any edge k-colouring of G there exists a vertex kcolouring of G in which the same colour never appears on an edge and both its endpoints. (Neither the edge nor the vertex colourings are necessarily proper in the usual sense.) We give an efficient characterization of graphs with adaptable chromatic number at most...

Let G be a simple graph with vertex set V(G) and edge set E(G). A subset S of V(G) is called an independent set if no two vertices of S are adjacent in G. The minimum number of independent sets which form a partition of V(G) is called chromatic number of G, denoted by χ(G). A subset S of E(G) is called an edge cover of G if the subgraph induced by S is a spanning subgraph of G. The maximum numb...

A dynamic k-coloring of a graphG is a proper k-coloring of the vertices ofG such that every vertex of degree at least 2 in G will be adjacent to vertices with at least two different colors. The smallest number k for which a graph G has a dynamic k-coloring is the dynamic chromatic number χd(G). In this paper, we investigate the behavior of χd(G), the bounds for χd(G), the comparison between χd(...

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets. In the spirit of a famous theorem of P. Erdős [Graph theory and probability, Canad. J. Math. 11 (1959), 34–38], it was shown prob...

In order to motivate our results we will begin by describing two classical expansions of the chromatic polynomial. Let G be a simple graph with finite vertex set V . Let χG(x) be the number of proper colorings of G with x colors (assignments of colors 1, 2, . . . , x to the vertices of G so that no two adjacent vertices have the same color). If e is an edge of G, let G\e be G with e removed, an...