Chromatic number, induced cycles, and non-separating cycles

TL;DR

This paper introduces new parameters based on induced and non-separating cycles to derive upper bounds on chromatic number and Hadwiger number, and provides improved lower bounds on induced non-separating cycles in 3-connected graphs.

Contribution

It establishes monotonicity of these parameters under certain graph operations and improves known bounds on cycles and graph parameters.

Findings

Derived new upper bounds on chromatic number and Hadwiger number.

Proved that 3-connected graphs with average degree k have at least (k-1)|V|+Ck^{3}log^{3/2}k induced non-separating cycles.

Provided a short proof of Tutte's cycle space theorem.

Abstract

We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge contraction, we obtain new upper bounds on the chromatic number in terms of the number of induced cycles and the Hadwiger number in terms of the number of induced non-separating cycles. As an application, we show that a 3-connected graph with average degree $k\ge 2$ have at least $(k-1)|V|+Ck^{3}\log^{3/2}k$ induced non-separating cycles for some explicit constant $C>0$. This improves the previous best lower bound $(k-1)|V|+1$, which follows from Tutte's cycle space theorem. We also give a short proof of this theorem of Tutte.