Injective edge-coloring of subcubic graphs

An injective edge-coloring [Formula: see text] of a graph [Formula: see text] is an edge-coloring such that if [Formula: see text], [Formula: see text], and [Formula: see text] are three consecutive edges in [Formula: see text] (they are consecutive if they form a path or a cycle of length three), then [Formula: see text] and [Formula: see text] receive different colors. The minimum integer [Formula: see text] such that, [Formula: see text] has an injective edge-coloring with [Formula: see text] colors, is called the injective chromatic index of [Formula: see text] ([Formula: see text]). This parameter was introduced by Cardoso et al. [Injective coloring of graphs, Filomat 33(19) (2019) 6411–6423, arXiv:1510.02626] motivated by the Packet Radio Network problem. They proved that computing [Formula: see text] of a graph [Formula: see text] is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. We study the injective edge-coloring of some classes of subcubic graphs. We prove that a subcubic bipartite graph has an injective chromatic index bounded by [Formula: see text]. We also prove that if [Formula: see text] is a subcubic graph with maximum average degree less than [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] admits an injective edge-coloring with at most 4 (respectively, [Formula: see text]) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs.

List injective coloring of planar graphs with disjoint 5−-cycles

An injective [Formula: see text]-coloring of a graph [Formula: see text] is called injective if any two vertices joined by a path of length two get different colors. A graph [Formula: see text] is injectively [Formula: see text]-choosable if for any color list [Formula: see text] of admissible colors on [Formula: see text] of size [Formula: see text] it allows an injective coloring [Formula: see text] such that [Formula: see text] whenever [Formula: see text]. Let [Formula: see text], [Formula: see text] denote the injective chromatic number and injective choosability number of [Formula: see text], respectively. Let [Formula: see text] be a plane with disjoint [Formula: see text]-cycles and maximum degree [Formula: see text]. We show that [Formula: see text] if [Formula: see text], then [Formula: see text]; [Formula: see text] if [Formula: see text], then [Formula: see text].

Injective coloring of complementary prism and generalized complementary prism graphs

The complementary prism [Formula: see text] of a graph [Formula: see text] is the graph obtained by drawing edges between the corresponding vertices of a graph [Formula: see text] and its complement [Formula: see text]. In this paper, we generalize the concept of complementary prisms of graphs and determine the injective chromatic number of generalized complementary prisms of graphs. We prove that for any simple graph [Formula: see text] of order [Formula: see text], [Formula: see text] and if [Formula: see text] is a graph with a universal vertex, then [Formula: see text].

List injective coloring of a class of planar graphs without short cycles

An injective [Formula: see text]-coloring of a graph [Formula: see text] is a mapping c: [Formula: see text]([Formula: see text]) [Formula: see text][Formula: see text] such that [Formula: see text] whenever [Formula: see text] have a common neighbor in [Formula: see text]. A list assignment of a graph [Formula: see text] is a mapping [Formula: see text] that assigns a color list [Formula: see text] to each vertex [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an injective coloring [Formula: see text] of [Formula: see text] is called an injective [Formula: see text]-coloring if [Formula: see text] for every [Formula: see text]. In this paper, we show that if [Formula: see text] is a planar graph with girth [Formula: see text], then [Formula: see text] if [Formula: see text].

A study on the injective coloring parameters of certain graphs

Coloring the vertices of a graph [Formula: see text] according to certain conditions can be considered as a random experiment and a discrete random variable (r.v.) [Formula: see text] can be defined as the number of vertices having a particular color in the proper coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. In this paper, we extend the concepts of mean and variance to a modified injective graph coloring and determine the values of these parameters for a number of standard graphs.