Acyclic coloring of graphs of maximum degree five: Nine colors are enough

Abstract

An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G. In this paper, we show that any graph of maximum degree 5 has acyclic chromatic number at most 9, and we give a linear time algorithm that achieves this bound.

Graphs with maximum degree 5 are acyclically 7-colorable

“…Kothapalli, Satish, and Venkaiah [11] proved that every graph with maximum degree r is acyclically colorable with at most 1 + r(3r + 4)/8 colors. This is better than the bound r(r − 1)/2 in [7] for r ≥ 8. The main result of this paper is Theorem 1.1.…”

“…Our proof is different from that in [7,9,12] and heavily uses the ideas of Burstein [6]: he started from an uncolored graph G with maximum degree 4 and colored step by step more and more vertices (with some recolorings) so that each of partial acyclic 5-colorings of G had additional good properties that enabled him to extend the coloring further. The proof yields a linear-time algorithm for acyclic coloring with at most 7 colors of any graph with maximum degree 5.…”

“…Fertin and Raspaud [7] showed among other results that a(5) ≤ 9 and gave a linear-time algorithm for acyclic 9-coloring of any graph with maximum degree 5. Furthermore, for every fixed r ≥ 3, they gave a fast algorithm that uses at most r(r − 1)/2 colors for acyclic coloring of any graph with maximum degree r. Of course, for large r this is much worse than the upper bound (1.1), but for r < 1000, it is better.…”

Universal Immersion Spaces for Edge-Colored Graphs and Nearest-Neighbor Metrics

“…It is slightly easier to obtain a universal immersion graph for edge-colored graphs with this definition, and Alon and Marshall [2] have done just this but prove more: they provide a universal immersion graph, in this weaker sense, for the set of edge-colored graphs whose edges are colored by c colors and which have acyclic chromatic number (see [6]) bounded by K. Their universal graph has cardinality bounded by Kc K−1 , which is almost tight. For large k it is known that the greatest possible acyclic chromatic number of graphs of maximum degree k is between Ω(k 4/3 / log 1/3 k) and O(k 4/3 ) (see [3], and see Fertin and Raspaud [5] The size of our construction:…”

Acyclic Vertex Coloring of Graphs of Maximum Degree Six

“…Burnstein [5] showed that a(G) ≤ 5 for any graph of degree maximum 4. The work of Skulrattanakulchai was extended by Fertin and Raspaud [6] to show that it is possible to acyclically vertex color a graph G of maximum degree Δ using at most Δ(Δ − 1)/2 colors. Recently, Yadav et al [10] extended the work of Skulrattanakulchai [9] to show that any graph of maximum degree 5 can be colored using at most 8 colors.…”