Subdivision Of Complete Graph Research Articles

Remarks on proper conflict-free colorings of graphs

A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χpcf(G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χpcf(G) in terms of other graph parameters. In particular, we show that χpcf(G)≤5Δ(G)/2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4≤k≤6. The paper concludes with few open problems.

A note on the metric and edge metric dimensions of 2-connected graphs

For a given graph G , the metric and edge metric dimensions of G , dim ( G ) and edim ( G ) , are the cardinalities of the smallest possible subsets of vertices in V ( G ) such that they uniquely identify the vertices and the edges of G , respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph G such that dim ( G ) = a and edim ( G ) = b for every pair of integers a , b , where 4 ≤ b < a . For this we use subdivisions of complete graphs, whose metric dimension is in some cases smaller than the edge metric dimension. Along the way, we present an upper bound for the metric and edge metric dimensions of subdivision graphs under some special conditions.