On conflict-free colorings of cyclic polytopes and the girth conjecture for graphs

Abstract

We study the conflict-free chromatic number of hypergraphs derived from the family of facets of $d$-dimensional cyclic polytopes with $n$ vertices. While in odd dimensions $d$ the problem is easy, for even dimensions the problem becomes very difficult and exhibits interesting connections to extremal graph theory. We provide sharp asymptotic bounds for the conflict-free chromatic number in several small even dimensions and non-trivial upper and lower bounds for general even dimensions. The main purpose of this paper is revealing a surprising relation between conflict-free colorings and the celebrated Erdős girth conjecture, opening new avenues for future research.

Figures (3)

• Figure 1: Illustration: A coloring on $[n]$ using $c$ colors from an Eulerian circuit $C_G$ of a graph $G$ with $c$ vertices and $n$ edges. For this example, we have $n=10$ and $c=5$, and the tour begins from $v_1$ and $1$ for $G$ and $C_{[10]}$, respectively. By the proof of Theorem \ref{['thm:tight_upper']} below, this also gives a CF-coloring of $\mathsf{FC}_4(10)$.
• Figure 2: Walecki's Hamiltonian path decomposition and illustration of a CF-coloring of $\mathsf{I}^2_n$ when $k=3$. For the rectangle $\mathop{\mathrm{conv}}\nolimits\{v_2,v_3, v_5, v_6\}$, the edge $v_2v_6$ and $v_3v_5$ are interior edges.
• Figure 3: (a) An $ABABA$-free ($(AB)^{2.5}$-free) hypergraph which is not a $2$-interval hypergraph. (b) $ABABAB$-free ($(AB)^{3}$-free) hypergraph which is not a sub-hypergraph of $\mathsf{H}^{\gamma_{4}}_{16}$ where $\gamma_4$ is described simply as a straight line. Each color describes distinct hyperedges. A similar construction can be obtained for more intervals or higher dimensions.

Theorems & Definitions (44)

• Theorem 1.2: Lower bounds on $\mathop{\mathrm{\chi_{cf}}}\nolimits(\mathsf{FC}_d(n))$ for even $d$
• Theorem 1.3: Sharp bounds on $\mathop{\mathrm{\chi_{cf}}}\nolimits(\mathsf{FC}_d(n))$ for small even $d$
• Theorem 1.4: Upper bounds for general even $d$
• Theorem 1.5
• Theorem 2.1: Gale's evenness criterion
• Proposition 2.2: Proper colorings
• proof
• Proposition 2.3: CF-colorings in odd dimensions
• proof
• proof : Proof of Theorem \ref{['thm:lower_bound_CF']} (1)