Recursive Euler and Hamilton Paths

Abstract

Abstract. We employ recursion theoretic arguments to show that Hamilton paths for locally finite graphs are more difficult to find, in general, than Euler paths. A locally finite graph is recursive if we can effectively decide whether or not any two given vertices are adjacent, and highly recursive if we can effectively find all vertices adjacent to any given vertex. We find that there are recursive planar graphs with Euler or Hamilton paths but no such recursive paths. There are even particularly simple classes of connected, planar, highly recursive graphs for which we can show there is no effective way to decide about the existence of Euler or Hamilton paths. However, we obtain the following contrast: If a highly recursive graph has an Euler path we can effectively find a recursive Euler path; whereas, there is a planar, highly recursive graph with Hamilton paths but no recursive Hamilton path.

Reverse Mathematics and Recursive Graph Theory

“…Now we will analyze the third and most difficult task. Theorem 30 is closely related to Harel's proof [7] that the problem of finding a Hamiltonian path is Σ 1 1 complete.…”

“…(Bounded graphs are defined in Section 3.) This result is just a formalization of Bean's[1] proof that every highly recursive pre-Eulerian graph has a recursive Euler path.Theorem 18 (RCA 0 ). If G is a bounded pre-Eulerian graph, then G has an Euler path.…”

“…Bean [1]. The formalization requires verification that Euler's Theorem for finite graphs (see [10]) can be proved using only RCA 0 .…”

“…These axiom systems, in strictly increasing order of strength, are Σ 1 1 −AC 0 , ATR 0 , and Π 1 1 −CA 0 . The subsystem Σ 1 1 -axiom of choice, denoted by Σ 1 1 −AC 0 , consists of RCA 0 together with the comprehension scheme…”

“…(1) Σ 1 1 −AC 0 . (2) If G i : i ∈ N is a sequence of graphs such that each G i has a Hamilton path, then there is a sequence P i : i ∈ N such that for each i, P i is a Hamilton path through…”