Acyclic Coloring Of Graphs Research Articles

Acyclic Coloring of Graphs with Maximum Degree 7

The acyclic chromatic number a(G) of a graph G is the minimum number of colors such that G has a proper vertex coloring and no bichromatic cycles. For a graph G with maximum degree $$\Delta $$ , Grunbaum (1973) conjectured $$a(G)\le \Delta +1$$ . Up to now, the conjecture has only been shown for $$\Delta \le 4$$ . In this paper, it is proved that $$a(G)\le 12$$ for $$\Delta =7$$ , thus improving the result $$a(G)\le 17$$ of Dieng et al. (in: Proc. European conference on combinatorics, graph theory and applications, 2010).

Acyclic coloring of graphs and entropy compression method

Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, Dujmović et al. (2016) initiated the use of the so-called entropy compression method for graph coloring problems. Then, using the same approach Esperet and Parreau (2013) proved new upper bounds for several chromatic numbers, and explained how that approach could be used for many different coloring problems. Here, we follow this line of research for the particular case of acyclic coloring: we show that every graph with maximum degree Δ has acyclic chromatic number at most 32Δ43+O(Δ).

Acyclic coloring of graphs with maximum degree at most six

An acyclic k-coloring of a graph G is a proper vertex coloring using k colors such that any cycle of G is colored with at least three colors, and the minimum integer k is called the acyclic chromatic number of G, denoted by χa(G). Grünbaum conjectured that the acyclic chromatic number of graphs with maximum degree at most Δ is no more than Δ+1 in 1973. By now, the conjecture has been proved only for Δ≤4. We show that χa(G)≤9 for Δ≤6, which improves the result χa(G)≤10 of Zhao et al. (2014).

Acyclic colorings of graphs with bounded degree

A $k$-colouring (not necessarily proper) of vertices of a graph is called {\it acyclic}, if for every pair of distinct colours $i$ and $j$ the subgraph induced by the edges whose endpoints have colours $i$ and $j$ is acyclic. In the paper we consider some generalised acyclic $k$-colourings, namely, we require that each colour class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic $5$-colouring such that each colour class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph $G$ has an acyclic 2-colouring in which each colour class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic $t$-improper colouring of any graph with maximum degree $d$ assuming that the number of colors is large enough.

Acyclic coloring of graphs without bichromatic long path

Let G be a graph of maximum degree Δ. A proper vertex coloring of G is acyclic if there is no bichromatic cycle. It was proved by Alon et al. [Acyclic coloring of graphs. Random Structures Algorithms, 1991, 2(3): 277–288] that G admits an acyclic coloring with O(Δ4/3) colors and a proper coloring with O(Δ(k-1)/(k-2)) colors such that no path with k vertices is bichromatic for a fixed integer k ≥ 5. In this paper, we combine above two colorings and show that if k ≥ 5 and G does not contain cycles of length 4, then G admits an acyclic coloring with O(Δ(k-1)/(k-2)) colors such that no path with k vertices is bichromatic.

Acyclic coloring of graphs with some girth restriction

A vertex coloring of a graph $$G$$G is called acyclic if it is a proper vertex coloring such that every cycle $$C$$C receives at least three colors. The acyclic chromatic number of $$G$$G is the least number of colors in an acyclic coloring of $$G$$G. We prove that acyclic chromatic number of any graph $$G$$G with maximum degree $$\Delta \ge 4$$Δ?4 and with girth at least $$4\Delta $$4Δ is at most $$12\Delta $$12Δ.

On acyclic colorings of graphs on surfaces

A properk-coloring of a graph is acyclic if every 2-chromatic subgraph is acyclic. Borodin showed that every planar graph has an acyclic 5-coloring. This paper shows that the acyclic chromatic number of the projective plane is at most 7. The acyclic chromatic number of an arbitrary surface with Euler characteristic η=−γ is at mostO(γ4/7). This is nearly tight; for every γ>0 there are graphs embeddable on surfaces of Euler characteristic −γ whose acyclic chromatic number is at least Ω(γ4/7/(logγ)1/7). Therefore, the conjecture of Borodin that the acyclic chromatic number of any surface but the plane is the same as its chromatic number is false for all surfaces with large γ (and may very well be false for all surfaces).

Acyclic coloring of graphs

AbstractA vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two‐colored cycle in G. The acyclic chromatic number of G, denoted by A(G), is the least number of colors in an acyclic coloring of G. We show that if G has maximum degree d, then A(G) = 0(d4/3) as d → ∞. This settles a problem of Erdös who conjectured, in 1976, that A(G) = o(d2) as d → ∞. We also show that there are graphs G with maximum degree d for which A(G) = Ω(d4/3/(log d)1/3); and that the edges of any graph with maximum degree d can be colored by 0(d) colors so that no two adjacent edges have the same color and there is no two‐colored cycle. All the proofs rely heavily on probabilistic arguments.