Eulerian circuits and path decompositions in quartic planar graphs

TL;DR

This paper characterizes quartic planar graphs that have Eulerian circuits avoiding small cycles and establishes conditions for decomposing such graphs into paths of specific lengths, including a universal result for even-order graphs.

Contribution

It provides a characterization of Eulerian circuits avoiding 3- and 4-cycles in quartic planar graphs and derives a decomposition theorem into paths of various lengths.

Findings

Quartic planar graphs avoiding 3- and 4-cycles are characterized.

Decomposition conditions for quartic planar graphs into paths are established.

Every connected even-order quartic planar graph admits a P5-decomposition.

Abstract

A subcycle of an Eulerian circuit is a sequence of edges that are consecutive in the circuit and form a cycle. We characterise the quartic planar graphs that admit Eulerian circuits avoiding 3-cycles and 4-cycles. From this, it follows that a quartic planar graph of order $n$ can be decomposed into $k_1+k_2+k_3+k_4$ many paths with $k_i$ copies of $P_{i+1}$, the path with $i$ edges, if and only if $k_1+2k_2+3k_3+4k_4 = 2n$. In particular, every connected quartic planar graph of even order admits a $P_5$-decomposition.