Published in Volume XXVI, Issue 2, 2016, pages 249–261, doi: 10.7561/SACS.2016.2.249

Authors: S. Saqaeeyan, E. Mollaahamdi

Abstract

A dynamic coloring of a graph G is a proper vertex coloring such that for every vertex
v Î V(G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest
integer k such that G has a dynamic coloring with k colors, is called the
dynamic chromatic number of G and denoted by c2(G).
Montgomery conjectured that for every r-regular graph G, c2(G)-c(G) ≤ 2 .
Finding an optimal upper bound for c2(G)-c(G) seems to be an intriguing problem.
We show that there is a constant d such that every bipartite graph G with d(G) ³ d , has c2(G)-c(G) ≤ 2é(D(G))/(d(G))ù.
It was shown that c2(G)-c(G) ≤ a’

(G) +k* . Also, c2(G)-c(G) ≤ a(G) +k* . We prove that if G is a simple graph with d(G)>2, then c2(G)-c(G) ≤ (a’

(G)+w(G) )/2 +k* .
Among other results, we prove that for a given bipartite graph G=[X,Y], determining whether G
has a dynamic 4-coloring l : V (G)®{a, b, c, d} such that a, b
are used for part X and c, d are used for part Y
is NP-complete.

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Bibtex

@article{sacscuza:saqaeeyan2016dcnobg,
  title={Dynamic Chromatic Number of Bipartite Graphs},
  author={S. Saqaeeyan and E. Mollaahamdi},
  journal={Scientific Annals of Computer Science},
  volume={26},
  number={2},
  organization={``A.I. Cuza'' University, Iasi, Romania},
  year={2016},
  pages={249–261},
  doi={ 10.7561/SACS.2016.2.249},
  publisher={``A.I. Cuza'' University Press}
}