Metric Chromatic Number of Unicyclic Graphs

Abstract

All graphs in this paper are nontrivial and connected graph. Let 𝑓 ∶ 𝑉 (𝐺) → *1,2, … , 𝑘+ be a vertex coloring of a graph 𝐺where two adjacent vertices may be colored the same color. Consider the color classes Π = *𝐶 , 𝐶 , … , 𝐶 +. For a vertex 𝑣of 𝐺, the representation color of 𝑣is the 𝑘-vector 𝑟(𝑣|Π) = (𝑑(𝑣, , 𝐶 ), 𝑑(𝑣, 𝐶 ), … , 𝑑(𝑣, 𝐶 )), where 𝑑(𝑣, 𝐶 ) = min *𝑑(𝑣, 𝑐); 𝑐 ∈ 𝐶 + . If 𝑟(𝑢|Π) ≠ 𝑟(𝑣|Π) for every two adjacent vertices 𝑢and 𝑣of 𝐺, then 𝑓is a metric coloring of 𝐺. The minimum 𝑘for which 𝐺has a metric 𝑘-coloring is called the metric chromatic number of 𝐺and is denoted by 𝜇(𝐺). The metric chromatic numbers of unicyclic graphs namely tadpole graphs, cycle with 𝑚-pendants, sun graphs, cycle with two pendants, subdivision of sun graphs.