A distinguishing coloring of a graph G is a coloring of the vertices so that every nontrivial automorphism of G maps some vertex to a vertex with a different color. The distinguishing number of G is the minimum k such that G has a distinguishing coloring where each vertex is assigned a color from {1, . . . , k}. A list assignment to G is an assignment L = {L(v)}v∈V (G) of lists of colors to the...

An edge-weighting of a graph is called vertex-coloring if the weighted degrees yield proper vertex coloring graph. It conjectured that for every without isolated edge, with set {1,2,3} exists. In this note, we show statement true weight {1,2,3,4}.

A k-edge-weighting w of a graph G is an assignment of an integer weight, w(e) ∈ {1, . . . , k}, to each edge e. An edge weighting naturally induces a vertex coloring c by defining c(u) = ∑ u∼e w(e) for every u ∈ V (G). A k-edge-weighting of a graph G is vertexcoloring if the induced coloring c is proper, i.e., c(u) ≠ c(v) for any edge uv ∈ E(G). Given a graph G and a vertex coloring c0, does th...

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...

An integer distance graph is a graph G(D) with the set Z of all integers as vertex set and two vertices u, v ∈ Z are adjacent if and only if |u− v| ∈ D, where the distance set D is a subset of positive integers. A k-vertex coloring of a graph G is a mapping f from V (G) to [0, k − 1]. A path k-vertex coloring of a graph G is a k-vertex coloring such that every connected component is a path in t...

Many classes of graphs where the vertex coloring problem is polynomially solvable are known, the most prominent being the class of perfect graphs. However, the list-coloring problem is NP-complete for many subclasses of perfect graphs. In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs, where the former admits polynomia...

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variab...

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with k colors. A simple graph G is called b-vertex (edge) critical if the removal of any vertex (edge) of G increases its b-chromatic number. In thi...

The list-chromatic number χ`(G) of a graph G = (V,E) is the least cardinal κ such that for every assignment of a list L(V ) of at least κ colors to each vertex v ∈ V there is a proper vertex coloring c of G assigning to each vertex v a color from its list L(V ). The coloring number of G is the least cardinal κ such that there exists a well-ordering ≺ of V with the property that |{u : {u, v} ∈ E...