Adjacent vertex distinguishing total coloring of graphs with maximum degree 4

Adjacent Vertex Distinguishing Total Coloring of Graphs with Lower Average Degree

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by χ′′ a(G). Let mad(G) and ∆(G) denote the maximum average degree and the maximum degree of a graph G, respectivel...

Adjacent vertex-distinguishing edge coloring of graphs

An adjacent vertex-distinguishing edge coloring, or avd-coloring, of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. Let mad(G) and ∆(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every graph G with ∆(G) ≥ 5 and mad(G) < 3− 2 ∆ can be avd-colored with ∆(G) + 1 col...

Acyclic Vertex Coloring of Graphs of Maximum Degree 4

The acyclic chromatic number of a graph G, denoted a(G), is the minimum number of colors required to properly color the vertices of a graph such that there are no bichromatic cycles. The concept of acyclic coloring of a graph was introduced by [5] and is further studied in the last two decades in several works. Kostochka [6] proves that determining it is an NP-complete problem. Given the comput...

Adjacent Vertex Distinguishing Total Coloring of Line and Splitting Graph of Some Graphs

adjacent vertex distinguishing acyclic edge coloring of the cartesian product of graphs

‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...