An improved upper bound on the adjacent vertex distinguishing chromatic index of a graph

Delta+300 is a bound on the adjacent vertex distinguishing edge chromatic number

The adjacent vertex-distinguishing total chromatic number of 1-tree

Let G = (V (G), E(G)) be a simple graph and T (G) be the set of vertices and edges of G. Let C be a k−color set. A (proper) total k−coloring f of G is a function f : T (G) −→ C such that no adjacent or incident elements of T (G) receive the same color. For any u ∈ V (G), denote C(u) = {f(u)} ∪ {f(uv)|uv ∈ E(G)}. The total k−coloring f of G is called the adjacent vertex-distinguishing if C(u) 6=...

An Upper Bound on the First Zagreb Index in Trees

In this paper we give sharp upper bounds on the Zagreb indices and characterize all trees achieving equality in these bounds. Also, we give lower bound on first Zagreb coindex of trees.

An improved bound on acyclic chromatic index of planar graphs

A proper edge coloring of a graph G is called acyclic if there is no bichromatic cycle in G. The acyclic chromatic index of G, denoted by χ′a(G), is the least number of colors k such that G has an acyclic edge k-coloring. Basavaraju et al. [Acyclic edgecoloring of planar graphs, SIAM J. Discrete Math. 25 (2) (2011), 463–478] showed that χ′a(G) ≤ ∆(G) + 12 for planar graphs G with maximum degree...

The relationship between the distinguishing number and the distinguishing index with detection number of a graph

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