# Euler path words.

The 26 capital letters of the alphabet can be topologically viewed as simple networks--collections of links joined by nodes. Nodes are classified by the number of links that meet there. A well-known theorem in graph theory states that a network contains an Euler Path (a path that traverses the network, once only along each link) if and only if it has at most two nodes with an odd number of links. (This theorem was once used by Euler to prove that one could not traverse the seven bridges of Konigsberg without repeating one or more of them.)If one posits a sans-serif alphabet, the Euler Path letters (ones that can be traced out without lifting pencil from paper) are BCDGIJLMNOPQSUVWZ. Words of Euler Path letters include

DISILLUSIONING

COMMISSIONING

CONDOMINIUMS

CUMULONIMBUS

CUNNILINGUISM

DISCOMMODING

DISCOMPOSING

DISCUSSIONS

DISILLUSIONS

DISINCLINING

IMMOBILIZING

INCOMMODIOUS

INCONSPICUOUS

LIBIDINIZING

MONOPOLIZING

NONCOMBINING

NONPOISONOUS

OSCILLOSCOPIC

SNOWMOBILING

SOLILOQUIZING

SUBCOMMISSION

SUBCONSCIOUS

UNCONVINCING

UNSUSPICIOUS

One can also construct words with no Euler Path letters, the longest being THEREAFTER.

AFFRAYER

AFTERTAX

EYETEETH

FEATHERY

FERRETER

FETTERER

HAFTARAH

HEATHERY

HETAERAE

REFERRER

REHEATER

TARTRATE

A strict Euler Path word is one in which one can trace all the letters without lifting pencil from paper and connecting successive letters with horizontal lines: CULM, CULP, ISIS, IWIS, JIMP, MILS, NIMS, NISI, SIMS, SIMP, SWIM, SWIMS, VIMS. WIMP.

The editor points out that Battus, on page hq of Opperlans! (Querido, 2002) has created a more detailed topological classification of the sans serif capital letters, adding endpoints to the mix. Euler Path words consist of the first four groups.

1. no nodes or endpoints: DO

2. no nodes, two endpoints: CGIJLMNSUVWZ

3. one three-node, one endpoint: PQ

4. two three-nodes: B

5. one three-node, three endpoints: EFTY

6. two three-nodes, two endpoints: AR

7. one four-node, four endpoints: KX

8. two three-nodes, four endpoints: H

Battus was unable to find a Dutch word containing one letter from each of his eight groups, so coined POCHBERK. There is an English word, however: BAKESHOP. Longer words containing all groups include HUMPBACKED, PHRASEBOOK, SHIPBROKER, XENOPHOBIA, and TRISKAIDEKAPHOBIA (fear of 13).

ERIC IVERSON

St Paul, Minnesota

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Author: | Iverson, Eric |
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Publication: | Word Ways |

Date: | Nov 1, 2004 |

Words: | 350 |

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