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A short proof of Meyniel's theorem
Discrete Mathematics 19 (1977) 195 .- 197 (CJ Nw th-Hofiand Publishing Company
A SHQRT PROOF OF MEYNIEC’S THEOREM
C. THOMASSENZ
Received 1X June 19%
Meynieli’s theorem states that a strict diconnected digraph has a directed Hamiitnn *ycle if - t for every pair u, u of nonadjacent vertices. We give a short prrlof of this d(r.4)+dC0)aZn theorem.
We use the terminology an%3riome of the notation of [I]. D will denote a digraph on n vertices with vertex set Y. A digraph is strict if it has no loops and no two arcs with the same ends have the same orientation, and &connected if, for any two vertices u and u, there is a directed path frorll u to L’. If u E V and S c V, we denote the set of arcs from u to S by E(o + S) and the set of arcs between t, and S by Ecu, S). An S-p& is a directed path of Iength at least two having exactly its zigin and tetminus in common with S. An x + y path is a directed path with origin x and terminus ye The following lemma is we&known. We include it for the sake of completeness. --+ vk be a directed path in a strict digraph n, and ket Lemma. Let P:vf-+vt-*v E V\ V(P). If D has no vl -+ vk path with vertex set V(P) U (t:), then pqu, V(P))/ s k + 1. Proof. For dach i, 1 G i G k - 1, we have:
Ur.irersiry
Matematisk
Waterloo, Ontatio. Canada. fnstitut, AarhIrs UJniversitet, Aarhus, Denmrk
sf Watcrfaa.
Theorem 1. Let D be a strict diconnec~ed digraph such thnc, for ,twry pair 11.4, v of nnnadjace nt vertices d(u)+
d(vj r,2n
- 1.
Then D contains Q directed Hamilto~l cycle.
Theorem
1 is a consequence
of Thtzotem 2 below.
be a str.ct dicsnnected nanhamihnian digraph and let xL + x 1 be a ditected cycle of LJ suck thut no directed cycle uf D car tains V( S ) as a pralper subset. Then D contahs a vertex v not in S, and tltere ais! numbsrs 9, p (1s (JLG k, 1 :Z p < k ) such that (i) the (arc from x, to v is present in D, (ii) v is adjacent to no x,+, with 1 6 i 6 /3, c;.nd (iii) d(v)+ d(x: C_p)4 2n - 1 - 0. 0
Theorem 2. Let S:&-_,J*-*“‘3
D
Proof. Suppose first there is 30 V(S)-path in D Then since D is diconnected and V(S) is a proper subset of V, D contains a directed cycle S ’ having precisely one vertex, say x,, in common with S. Let u denote the successof of X, in S’. If 0 contains a path of the form x~+~-’ y 4 u or t 4 y 4 x~+~, where y E V\ V(S). then clearly we get a contradication to the assumption that D has no V(S)-path. So we can assume that no such path exists. Also, we can assume that v is adjacent to nc vertex of S other than x,. Hence d(v)+d(.r,+J<2+2(k
- 1) +
2
E(y, ka*lr v,‘>f
yEV’IV(S) 6
2k t 2(n - k - 1) = 2n - 2,
and the theorem is proved with ~3= 1. that D contains a V(S)-path f7 : X, -+ ye-+ y2+ Suppose thereiore p be chosen SO that y is minimum. path 3 y, 3 &acy* L-et the Because of the maximality property of S, y > 1. Put 19= yl. By the maximalify property of S, and by the lemma, we have: (1) u is joined to the path x,+, 4 x, +,,ef 4 + x by at most k - y + 2 edges. Because of the minimality of y, we get: (2) L‘ is not adjacent to any x,+,, with 14 i < J(, and D contains no path of the fsrm x,,, 4 y 4v, or v 4 y -4x,+,, with yEVV(S),and lsi
l
d(r.,)+d(x,+,)s(k +
c
-y
l
+p + 1>+2(y 93 - l)+
!E(y,(v,x,.,})J~2t;-p+1+2(~-k-1)=2n-l-~.
yEVIV(S) T%s
-y +2)+(&
l
proves the theorem.
l
19-?
“ References i d
! fll J.A. Bandy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London. 1076). yi [2] M. Meyniel, Une condition &&ante d’existence d’un circuit h,uniitonien dans un graph: orientd, I k$ Combinatorial Theory Ser. B 14 (lW3) 137-147. 15: tc [3] M. Overbeck-Larisch, Hamiftonian paths in oriented graphs, J. Combinatorial Theory Ser. R 2:
A SHQRT PROOF OF MEYNIEC’S THEOREM
C. THOMASSENZ
Received 1X June 19%
Meynieli’s theorem states that a strict diconnected digraph has a directed Hamiitnn *ycle if - t for every pair u, u of nonadjacent vertices. We give a short prrlof of this d(r.4)+dC0)aZn theorem.
We use the terminology an%3riome of the notation of [I]. D will denote a digraph on n vertices with vertex set Y. A digraph is strict if it has no loops and no two arcs with the same ends have the same orientation, and &connected if, for any two vertices u and u, there is a directed path frorll u to L’. If u E V and S c V, we denote the set of arcs from u to S by E(o + S) and the set of arcs between t, and S by Ecu, S). An S-p& is a directed path of Iength at least two having exactly its zigin and tetminus in common with S. An x + y path is a directed path with origin x and terminus ye The following lemma is we&known. We include it for the sake of completeness. --+ vk be a directed path in a strict digraph n, and ket Lemma. Let P:vf-+vt-*v E V\ V(P). If D has no vl -+ vk path with vertex set V(P) U (t:), then pqu, V(P))/ s k + 1. Proof. For dach i, 1 G i G k - 1, we have:
Ur.irersiry
Matematisk
Waterloo, Ontatio. Canada. fnstitut, AarhIrs UJniversitet, Aarhus, Denmrk
sf Watcrfaa.
Theorem 1. Let D be a strict diconnec~ed digraph such thnc, for ,twry pair 11.4, v of nnnadjace nt vertices d(u)+
d(vj r,2n
- 1.
Then D contains Q directed Hamilto~l cycle.
Theorem
1 is a consequence
of Thtzotem 2 below.
be a str.ct dicsnnected nanhamihnian digraph and let xL + x 1 be a ditected cycle of LJ suck thut no directed cycle uf D car tains V( S ) as a pralper subset. Then D contahs a vertex v not in S, and tltere ais! numbsrs 9, p (1s (JLG k, 1 :Z p < k ) such that (i) the (arc from x, to v is present in D, (ii) v is adjacent to no x,+, with 1 6 i 6 /3, c;.nd (iii) d(v)+ d(x: C_p)4 2n - 1 - 0. 0
Theorem 2. Let S:&-_,J*-*“‘3
D
Proof. Suppose first there is 30 V(S)-path in D Then since D is diconnected and V(S) is a proper subset of V, D contains a directed cycle S ’ having precisely one vertex, say x,, in common with S. Let u denote the successof of X, in S’. If 0 contains a path of the form x~+~-’ y 4 u or t 4 y 4 x~+~, where y E V\ V(S). then clearly we get a contradication to the assumption that D has no V(S)-path. So we can assume that no such path exists. Also, we can assume that v is adjacent to nc vertex of S other than x,. Hence d(v)+d(.r,+J<2+2(k
- 1) +
2
E(y, ka*lr v,‘>f
yEV’IV(S) 6
2k t 2(n - k - 1) = 2n - 2,
and the theorem is proved with ~3= 1. that D contains a V(S)-path f7 : X, -+ ye-+ y2+ Suppose thereiore p be chosen SO that y is minimum. path 3 y, 3 &acy* L-et the Because of the maximality property of S, y > 1. Put 19= yl. By the maximalify property of S, and by the lemma, we have: (1) u is joined to the path x,+, 4 x, +,,ef 4 + x by at most k - y + 2 edges. Because of the minimality of y, we get: (2) L‘ is not adjacent to any x,+,, with 14 i < J(, and D contains no path of the fsrm x,,, 4 y 4v, or v 4 y -4x,+,, with yEVV(S),and lsi
l
d(r.,)+d(x,+,)s(k +
c
-y
l
+p + 1>+2(y 93 - l)+
!E(y,(v,x,.,})J~2t;-p+1+2(~-k-1)=2n-l-~.
yEVIV(S) T%s
-y +2)+(&
l
proves the theorem.
l
19-?
“ References i d
! fll J.A. Bandy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London. 1076). yi [2] M. Meyniel, Une condition &&ante d’existence d’un circuit h,uniitonien dans un graph: orientd, I k$ Combinatorial Theory Ser. B 14 (lW3) 137-147. 15: tc [3] M. Overbeck-Larisch, Hamiftonian paths in oriented graphs, J. Combinatorial Theory Ser. R 2: