Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number · arXivDesk
0911.4199 Nov 21, 2009 Dynamic chromatic number; conditional (k, 2)-coloring; 2-hued chromatic number; 2-hued coloring; Independence number; Probabilistic method
Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number Arash Ahadi, Ali Dehghan
Abstract
A 2-hued coloring of a graph G G G (also known as conditional ( k , 2 ) (k, 2) ( k , 2 ) -coloring and dynamic coloring) is a coloring such that for every vertex v ∈ V ( G ) v\in V(G) v ∈ V ( G ) of degree at least 2 2 2 , the neighbors of v v v
receive at least
colors. The smallest integer
such that
has a 2-hued coloring with
colors, is called the 2-hued chromatic number of
and denoted by
. In this paper, we will show that if
is a regular graph, then
χ 2 ( G ) − χ ( G ) ≤ 2 log 2 ( α ( G ) ) + O ( 1 ) χ_{2}(G)- χ(G) \leq 2 \log _{2}(α(G)) +\mathcal{O}(1) χ 2 ( G ) − χ ( G ) ≤ 2 log 2 ( α ( G )) + O ( 1 ) and if
is a graph and
, then
χ 2 ( G ) − χ ( G ) ≤ 1 + ⌈ 4 Δ 2 δ − 1 ⌉ ( 1 + log 2 Δ ( G ) 2 Δ ( G ) − δ ( G ) ( α ( G ) ) ) χ_{2}(G)- χ(G) \leq 1+\lceil \sqrt[δ-1]{4Δ^{2}} \rceil ( 1+ \log _{\frac{2Δ(G)}{2Δ(G)-δ(G)}} (α(G)) ) χ 2 ( G ) − χ ( G ) ≤ 1 + ⌈ δ − 1 4 Δ 2 ⌉ ( 1 + log 2Δ ( G ) − δ ( G ) 2Δ ( G ) ( α ( G ))) and in general case if
is a graph, then
χ 2 ( G ) − χ ( G ) ≤ 2 + min { α ′ ( G ) , α ( G ) + ω ( G ) 2 } χ_{2}(G)- χ(G) \leq 2+ \min \lbrace α^{\prime}(G),\frac{α(G)+ω(G)}{2}\rbrace χ 2 ( G ) − χ ( G ) ≤ 2 + min { α ′ ( G ) , 2 α ( G ) + ω ( G ) } .