Aequationes Mathematicae 22 (1981) 42--45 University of Waterloo

0001-9054/81/010042-04501.50+0.20/0 © 1981 Birkh~iuser Verlag, Basel

Edge-disjoint Hamilton cycles in 4-regular planar graphs

J. A. BONDY AND R. HAOGKVIST

The problem of determining which 4-regular graphs are decomposable into two edge-disjoint Hamilton cycles was first considered by Kotzig [4], who proved that a 3-regular graph is hamiltonian if and only if its edge graph (that is, line graph) has a hamiltonian decomposition. With the aid of this theorem, it is an easy matter to construct examples of 4-regular graphs which fail to have such a decomposition.

Later, and presumably unaware of Kotzig's work (which was published in Slovak), Nash-Williams [7] conjectured that every 4-connected 4-regular graph admits a hamiltonian decomposition. Meredith [6] disproved this conjecture by constructing a nonhamiltonian 4-connected 4-regular graph. Gr/inbaum and Zaks [3] then proposed a weaker version of Nash-Williams' conjecture, restricted to planar graphs. (Recall that, by a theorem of Tutte [8], all 4-connected planar graphs are hamiltonian.) Counterexamples to this weaker conjecture were found by Gr/inbaum and Malkevitch [2] and also by Martin [5], who independently rediscovered Kotzig's theorem.

The purpose of this note is to point out that Grinberg's necessary condition for a plane graph to be hamiltonian [1] can be used to derive a similar necessary condition for a 4-regular plane graph to admit a hamiltonian decomposition. To simplify the statement of Grinberg's theorem and its subsequent application, we make the following definition. If G is any plane graph with face set F, let g : 2 ~ ~ N be defined by

g(X) = ~ ( d ( f ) - 2 ) f~x

where d(f) is the degree of f in G (that is, the number of edges in its boundary).

G R I N B E R G ' S T H E O R E M . Let G be a plane graph which contains a Hamil- ton cycle C. Denote by F1 the set of faces of G interior to C and by 1:;2 the set of

AMS (1980) subject classification: Primary 05C10, 05C40. This research was supported by a grant from the National Research Council of Canada.

Manuscript received October 27, 1978.

42

Vol. 22, 1981 Edge-disjoint Hamilton cycles 43

faces exterior to C. Then

g(F0 = g(r~). (1)

Thus, a necessary condition for a plane graph to be hamiltonian is that its face set F admit a partition (F1, F2) satisfying (1).

In the statement of the following theorem, we shall use the fact that every 4-regular plane graph, being eulerian, has a (unique) 2-face colouring.

THEOREM. Let G be a 4-regular plane graph which is decomposable into two edge-disjoint Hamilton cycles C and D. Denote by F1x, F12, F21 and F22 the sets of faces of G interior to both C and D, interior to C but exterior to D, interior to D but exterior to C, and exterior to both C and D, respectively. Then

g(F,,) = g(F22) (2)

and

g(F,2) = g(F2,). (3)

Thus, a necessary condition for a 4-regular plane graph to have a hamittonian decomposition is that the two color-classes of faces admit partitions (Fll, F22) and (F12, F20 satisfying (2) and (3), respectively.

Proof. Applying (1) to C and D, we obtain

g(FH) + g(F12) = g(F,, U F12 ) = g(F211,3 F22 ) = g(f21 ) + g(f22 )

g(Fll) + g(f21) = g(FH tO F21 ) = g(F,2 U F22 ) = g(F~2 ) + g(F22 ).

These two equations immediately yield (2) and (3).

The medial graph M(G) of a plane graph G is the graph with vertex set E(G) in which two vertices are joined by k edges (where k = 0, 1 or 2) if, and only if, in G, they are adjacent edges and are incident with k common faces. Every medial graph M(G) is 4-regular, with one colour-class of faces corresponding to the faces of G and the other colour-class corresponding to the faces of G*, the dual of G. (Conversely, every 4-regular plane graph is the common medial graph of two dual graphs.) Since this correspondence between faces is degree-preserving, we have the following corollary.

44 J. A. B O N D Y A N D R. H A G G K V I S T AEQ. MATH.

<Y G M(G)

Figure 1 The Herschel graph G and its medial graph M(G).

(G nonhamiltonian, G* hamiltonian, M(G) indecomposable by corollary).

C O R O L L A R Y . I f G is a plane graph and either G or G* fails to satisfy Grinberg's criterion, then M(G) does not admit a hamiltonian decomposition.

This corollary shows, for example, that the medial graph of the Herschel graph (Fig. 1) is not decomposable into two edge-disjoint Hamilton cycles, a fact noted by Griinbaum and Malkevitch [2, Fig. 1] using ad hoc methods.

R E M A R K S A N D PROBLEMS. In view of Kotzig's theorem, it is natural to ask whether the decomposability of a 4-regular plane graph is dependent on the hamiltonian characters of its associated dual graphs. The examples given in Figs. 1 and 2 show that no strong connection exists. However, we know of no indecom- posable 4-regular plane graph both of whose associated graphs are hamiltonian.

G

Figure 2 The Reynolds graph G.

(G nonhamiltonian, G* hamiltonian, M(G) decomposable by Kotzig's theorem).

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REFERENCES

[1] GRINnERG, E., Plane homogeneous graphs of degree three without Hamiltonian circuits. (Russian. Latvian and English summaries) Latvian Math. Yearbook, Izdat. "Zinatne", Riga 4 (1968), 51-58.

[2] GRONBAUM, B. and /VlALKEvrrcH, J., Pairs of edge-disjoint Hamiltonian circuits. Aequationes Math. 14 (1976), 191-196.

[3] GRONSAUM, B. and ZAKS, J., The existence of certain planar maps. Discrete Math. 10 (1974), 93--t15.

[4] Ko'rzm, A., Aus dec Theorie dec endIichen regulglren Graphen dritten und vierten Grades. (Slovak. Russian and German summaries) Casopis P~st. Mat. 82 (1957), 76-92.

[5] MARTIN, P., Cycles Hamiltoniens darts les graphes 4-r~guliers 4-connexes. Aequationes Math. 14 (1976), 37-40.

[6] MEREDITH, G. H. J., Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs. J. Combinatorial Theory Set. B 14 (1973), 55-60.

[7] NASH-WILLIAMS, C. ST. J. A,, Possible directions in graph theory. In: Proceedings of conference on Combinatorial Mathematics and its Applications, Oxford, (1969). Academic Press, London, I97 I, pp. 191-200.

[8] TtrrrE, W. T., A theorem on planar graphs. Trans. Amer. Math. Soc. 82 (1956), 99-116.

Dept. of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1