Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1705.05023

Acyclic edge-coloring of planar graphs: $Δ$ colors suffice when $Δ$ is large

Daniel W. Cranston
View original source ↗Read source document ↗Export metadata ↓

Abstract

An \emph{acyclic edge-coloring} of a graph $G$ is a proper edge-coloring of $G$ such that the subgraph induced by any two color classes is acyclic. The \emph{acyclic chromatic index}, $χ'_a(G)$, is the smallest number of colors allowing an acyclic edge-coloring of $G$. Clearly $χ'_a(G)\ge Δ(G)$ for every graph $G$. Cohen, Havet, and Müller conjectured that there exists a constant $M$ such that every planar graph with $Δ(G)\ge M$ has $χ'_a(G)=Δ(G)$. We prove this conjecture.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniel W. Cranston. 2019-01-29. Acyclic edge-coloring of planar graphs: $Δ$ colors suffice when $Δ$ is large. https://arxiv.org/abs/1705.05023

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗