Vol. 27, No. 2, 1968

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On interchange graphs

Benjamin L. Schwartz

Vol. 27 (1968), No. 2, 393–396
Abstract

The interchange graph I(G) for an unoriented graph G has been defined by Ore as follows: The vertices of I(G) are the edges of G; and two vertices of I(G) are connected by an edge if and only if they are adjacent (i.e., have a vertex in common) in G. In 1962 Ore raised the problem of determining those graphs for which

I(G ) = G.
(1)

This paper solves this problem for finite connected graphs with loops and parallel edges, extending earlier work on the problem.

A loop (an edge whose two ends coincide in a single vertex) is considered adjacent to itself, and hence generates another loop under the I mapping. If two edges of G connect the same two vertices, the corresponding vertices of I(G) are also connected by two distinct edges.

Mathematical Subject Classification
Primary: 05.40
Milestones
Received: 26 June 1967
Published: 1 November 1968
Authors
Benjamin L. Schwartz