In this paper we prove that if G is a (k + 2)-connected graph on n &gt; 3 vertices satisfying P(n + k) : dG(x; y) = 2 ) maxfd(x); d(y)g &gt; n + k 2 for each pair of vertices x and y in G; then any path S  G of length k is contained in a hamiltonian cycle of G

GANCARZEWICZ, GRZEGORZ

Portal Czasopism Naukowych (E-Journals)

GRZEGORZ GANCARZEWICZ∗
GRAPHS WITH EVERY PATH OF LENGTH k IN A
HAMILTONIAN CYCLE
GRAFY Z DOWOLNĄ ŚCIEŻKĄ DŁUGOŚCI k ZAWARTĄ
W PEWNYM CYKLU HAMILTONOWSKIM
Abs t r a c t
In this paper we prove that if G is a (k + 2)-connected graph on n > 3 vertices satisfying
P(n+ k) :
dG(x, y) = 2 ⇒ max{d(x), d(y)} > n+ k
2
for each pair of vertices x and y in G, then any path S ⊂ G of length k is contained in a
hamiltonian cycle of G.
Keywords: cycle, graph, hamiltonian cycle, matching, path
S t r e s z c z e n i e
W pracy udowodniono, że W (k + 2)-spójnym grafie G o n > 3 wierzchołkach, który spełnia
warunek P(n+ k) :
d(x, y) = 2 ⇒ max{d(x), d(y)} > n+ k
2
dla dowolnej pary wierzchołków x i y, każda ścieżka S ⊂ G długości k jest zawrta w pewnym
cyklu hamiltonowskim grafu G.
Słowa kluczowe: cykl, cykl hamiltonowski, graf, skojarzenie, ścieżka
∗Institute of Mathematics, Cracow University of Technology; ggancarzewicz@gmail.com
46
1. Introduction
We consider only finite graphs without loops and multiple edges. By V or V(G) we
denote the vertex set of the graph G and respectively by E or E(G), the edge set of
G. By dx(G) or d(x), we denote the degree of a vertex x in the graph G and by d(x, y)
or dG(x, y), the distance between x and y in G.
Definition 1.1 (cf [10]). Let k, s1, . . . s` be positive integers. We call S a path system
of length k, if the connected components of S are paths:
P 1 : x10x
1
1 . . . x
1
s1 ,
...
P l : x`0x
l
1 . . . x
`
s`
and
∑`
i=1 si = k .
Let S be a path system of length k and let x ∈ V(S) . We shall call x an internal
vertex if x is an internal vertex (cf [3]) in one of the paths P 1 , . . . , P ` .
If q denotes the number of internal vertices in a path system S of length k then
0 6 q 6 k − 1 . If q = 0, then S is a k-matching (i.e. a set of k independent edges).
Let H be a subgraph of G. By G \H we denote the graph obtained from G by the
deletion of the edges of H.
Definition 1.2. The graph F is said to be an H-edge cut-set of G if F ⊂ E(H) and
G \ F is not connected.
Definition 1.3. The graph F is said to be a minimal H-edge cut-set of G if F is an
H-edge cut-set of G which has no proper subset being an edge cut-set of G.
Definition 1.4 (cf [7]). Let G be a graph on n > 3 vertices and k > 0. G is k-edge-
hamiltonian if for every path system P of length at most k there exists a hamiltonian
cycle of G containing P.
Let G be a graph and H ⊂ G a subgraph of G. For a vertex x ∈ V(G), we define
the set NH(x) = {y ∈ V(H) : xy ∈ E(G)} . Let H and D be two subgraphs of G.
E(D,H) = {xy ∈ E(G) : x ∈ V(D) and y ∈ V(H)}. For a set of vertices A of a graph
G, we define the graph G(A) as the subgraph induced in G by A. In the proof, we
will only use oriented cycles and paths. Let C be a cycle and x ∈ V (C) , then x− is
the predecessor of x and x+ is its successor.
Definition 1.5 (cf [2]). Let W be a property defined for all graphs of order n and let
k be a non-negative integer. The property W is said to be k-stable if whenever G+ xy
has property W and d(x) + d(y) > k then G itself has property W.
47
J.A. Bondy and V. Chvátal [2] proved the following theorem, which we shall need
in the proof of our main result:
Theorem 1.1. Let n and k be positive integers with k 6 n− 3 . Then the property
of being k-edge-hamiltonian is (n+ k)-stable.
In 1960, O. Ore [9] proved the following:
Theorem 1.2. Let G be a graph on n > 3 vertices. If for all nonadjacent vertices
x, y ∈ V(G) we have
d(x) + d(y) > n
then G is hamiltonian.
Geng-Hua Fan [4] has shown:
Theorem 1.3. Let G be a 2-connected graph on n > 3 vertices. If G satisfies
P(n) : d(x, y) = 2⇒ max{d(x),d(y)} > n
2
for each pair of vertices x and y in G, then G is hamiltonian.
The condition for degree sum in Theorem 1.2 is called an Ore condition or an Ore
type condition for graph G and the condition P(k) is called a Fan condition or a Fan
type condition for graph G.
Later, many Fan type theorems and Ore type theorems are shown.
Now we shall present Las Vergnas [8] condition Ln,s .
Definition 1.6. Let G be graph on n > 2 vertices and let s be an integer such
that 0 6 s 6 n. G satisfies Las Vergnas condition Ln,s if there is an arrangement
x1 , . . . , xn of vertices of G such that for all i, j if
1 6 i < j 6 n , i+ j > n− s , xixj 6∈ E (G) ,
d(xi) 6 i+ s and d(xj) 6 j + s− 1
then d(xi) + d(xj) > n+ s.
Las Vergnas [8] proved the following theorem:
Theorem 1.4. Let G be a graph on n > 3 vertices and let 0 6 s 6 n − 1. If G
satisfies Ln,s then G is s-edge hamiltonian.
Note that condition Ln,s is weaker than Ore condition.
Later Skupień and Wojda proved that the condition Ln,s is sufficient for a graph
to have a stronger property (for details see [10]). Wojda [11] proved the following Ore
type theorem:
48
Theorem 1.5. Let G be a graph on n > 3 vertices, such that for every pair of
nonadjacent vertices x and y
d(x) + d(y) >
4n− 4
3
.
Then every matching of G lies in a hamiltonian cycle.
In 1996, G. Gancarzewicz and A. P. Wojda proved the following Fan type theorem:
Theorem 1.6. Let G be a 3-connected graph of order n > 3 and let M be a
k-matching in G. If G satisfies P(n+ k) :
d(x, y) = 2 ⇒ max{d(x),d(y)} > n+ k
2
for each pair of vertices x and y in G, then M lies in a hamiltonian cycle of G or G
has a minimal odd M -edge cut-set.
In this paper we find a Fan type condition under which every path of length k in
a graph G lies in a hamiltonian cycle.
For notation and terminology not defined above a good reference should be [3].
2.Result
Theorem 2.1. Let G be a graph on n > 3 vertices and let S be a path of length
k in G. If the graph G is l-connected, where l = min{k + 2 , n − 1} and satisfies
P(n+ k) :
d(x, y) = 2 ⇒ max{d(x),d(y)} > n+ k
2
(2.1)
for each pair of vertices x and y ∈ V(G), then S lies in a hamiltonian cycle of G.
Note that 1 6 k 6 n − 1 and since an (n − 1)-connected graph of order n is a
complete graph Kn, which is obviously k-edge hamiltonian for any k the result is
interesting when k < n− 3.
For k = 1, the path S is a 1-matching and we have a special case of Theorem 1.6
(the graph is 3-connected, so in this case we can not have a minimal S-edge cut set).
Unfortunately, in Theorem 2.1 we can not decrease the connectivity of graph G.
We can consider a vertex x and a complete graph Km , m > 3 . In the complete graph
Km we choose a path S : s1 . . . sk+1 of length k = m − 2 . There is only one vertex
y ∈ Km not contained in S.
Let G be a graph of order n = m+ 1 obtained from two complete graphs K1 = {x}
and Km by adding edges xsi , for i ∈ {1, . . . , k + 1} The path S is a path of length k
contained in G which is not contained in any hamiltonian cycle of the (k+1)-connected
graph G, see Figure (1).
49
x
Km
s1 s2 sk sk+1
y
G
edge from the path S.
Fig. 1: A (k + 1)-connected graph G with no hamiltonian cycle through the path S :
s1 . . . sk+1.
Note that we can replace the vertex x by a complete graph K` , ` > k + 1 . Let
{x1 , . . . , xk+1} ⊂ V(K`) and let G be a graph of order n = m+ l obtained from two
complete graphs K` and Km by adding edges xisi , for i ∈ {1, . . . , k + 1} The path S
is a path of length k contained in G which is not contained in any hamiltonian cycle
of the (k + 1)-connected graph G, see Figure (2).
3. Proof
Proof of Theorem 2.1:
Take G and S as in the assumptions of Theorem 2.1.
Consider the nonempty set
A = {x ∈ V (G) : dx(G) > n+ k
2
} .
Note that if x and y are nonadjacent vertices of A, then the graph obtained from G
by the addition of the edge xy also satisfies the assumptions of the theorem. Therefore,
and by Theorem 1.1 we may assume that:
xy ∈ E(G) for any x , y ∈ A and x 6= y . (3.1)
By (3.1), A induces a complete subgraph G(A) of the graph G.
In fact, since the property of being k-edge-hamiltonian is (n+ k)-stable, we can
replace G with its (n+ k)-closure.
Let GA be a graph obtained from G by deletion of vertices of the graph G(A) (i.e.
vertices from the set A).
Now take D, a connected component of the graph GA.
50
x1 x2 xk xk+1
Kℓ
Km
s1 s2 sk sk+1
y
G
edge from the path S.
Fig. 2: A (k + 1)-connected graph G with no hamiltonian cycle through the path S :
s1 . . . sk+1.
Suppose that there exist two nonadjacent vertices in D. Since D is connected, we
have two vertices x and y in D such that dG(x, y) = 2 and by the assumption that G
satisfies P(n+ k), we have x ∈ A or y ∈ A, which is a contradiction.
We have proved that every component of GA is a complete graph Kι , ι ∈ I , joined
with G(A) by at least k + 2 edges.
Claim 3.1. If Kι0 ,Kι1 ∈ {Kι}ι∈I are such that ι0 6= ι1, then:
N(Kι0) ∩N(Kι1) = ∅. (3.2)
Proof of Claim 3.1:
Suppose that N(Kι0) ∩N(Kι1) 6= ∅. Then we have a vertex y ∈ Kι0 and a vertex
y′ ∈ Kι1 such that dG(y, y′) = 2 and by P(n + k) either y ∈ A or y′ ∈ A . This
contradicts the fact that Kι0 and Kι1 are two connected components of GA .

We have shown that the graph G consists of a complete graph G(A) and of a family
of complete components {Kι}ι∈I , of GA , which do not have common neighbors in
G(A).
Since G is (k + 2)-connected, we have the following:
Claim 3.2. Every component {Kι}ι∈I , is joined with G(A) by at least three edges
such that end vertices of these edges are not internal vertices of the path S.
51
We label vertices of path S : s1s2 . . . sksk+1 .
Graph G consists of complete graph G(A) and disjoined complete graphs {Kι}ι∈I ,
joined with G(A) by at least three edges such that end vertices of these edges are not
internal vertices of path S.
Firstly we consider the case when path S is contained in one complete graph (i.e.
G(A) or one graph Kι0 ∈ {Kι}ι∈I). In this case, by Claim3.2 we have a hamiltonian
cycle through S.
Now we assume that S is not contained in the complete graph G(A) or one graph
Kι0 ∈ {Kι}ι∈I and we can now define a cycle C ⊂ G containing the path S and all
vertices of G(A).
We shall consider four cases:
1. Both end vertices of S are in G(A) i.e. s1 , sk+1 ∈ G(A).
2. Both end vertices of S are in the same component Kι of GA i.e. s1 , sk+1 ∈ Kι.
3. End vertices of S are in different components of GA i.e. s1 ∈ Kι1 , sk+1 ∈ Kι2 ,
Kι1 , Kι2 ∈ {Kι}ι∈I are such that ι1 6= ι2.
4. One end vertex of S is in G(A) and the other end vertex is in a component Kι
of GA. In this case, we can assume without loss of generality that s1 ∈ G(A)
and sk+1 ∈ Kι.
If C ⊂ G is a cycle in G, then by GV \ C we denote a graph obtained from G by
deletion of vertices of cycle C.
Case 1: Both end vertices of S are in G(A) i.e. s1 , sk+1 ∈ G(A).
Note that even in this case, path S may pass through some components Kι creating
a kind of ears of the complete graph G(A), on every incident graph Kι. We can find
an example of such ears on Figure (3).
Since G(A) is a complete graph we have a cycle C containing the path S and all
vertices of G(A) performing the following conditions:
52
Kι
G(A)
edge from the path S.
Fig. 3: Example of ears of the graph G(A).
• C contains all edges of E(S) ∩ E(G(A)) and all vertices of A . (3.3)
• If Kι0 and Kι1are two different components of GV \ C then (3.4)
N(Kι0) ∩N(Kι1) = ∅ .
• Let x 6∈ V (C) , y ∈ V (C) and xy ∈ E(G) then: (3.5)
if y is not an internal vertex of S , then y ∈ A,
if y− is not an internal vertex of S , then y− ∈ A,
if y+ is not an internal vertex of S , then y+ ∈ A.
Properties (3.3 — 3.5) will allow us to extend C to a hamiltonian cycle.
Note that this cycle C may not be contained in G(A).
Subcase 1.1: Extending the cycle C through components Kι incident with
ears
Since graph G is (k + 2)-connected, we have at least k + 2 edges joining Kι with
G(A), so at least one of these edges say uci , u ∈ V(Kι) , ci ∈ V(G(A)) \ S , is not
incident with S. If a component Kι is incident with an ear, at least one interior vertex
of path S is contained in this ear, and we have an additional edge say u′cj , u′ ∈ V(Kι) ,
cj ∈ V(G(A)) \ S , not incident with S joining Kι with G(A). Using these two edges
uci and u′cj , we can extend the cycle C through the remaining vertices of Kι.
Without loss of generality, we can assume that on cycle C, the vertices are ordered
in the following way: s1 . . . sk+1ck+2 . . . ci . . . cj . . . s1 .
53
ci
u u′
cjc−jc+i
c+j
Kι
C
s1 s2 sk sk+1s+k+1
P (u, u′)
edge from the path S.
Fig. 4: Extension of the cycle C through component Kι incident with an ear.
We can replace cycle C, by the following cycle C ′
C ′ : s1 . . . sk+1s+k+1 . . . ciP (u, u
′)cjc−j . . . c
+
i c
+
j . . . s1 ,
where P (u, u′) ⊂ Kι is a path joining u with u′ containing all vertices of Kι \ S.
Note that this cycle C ′ satisfies conditions (3.3 — 3.5). We can find an example of
the cycle C ′ on Figure 4.
Case 2: Both end vertices of S are in the same component Kι of GA i.e.
s1 , sk+1 ∈ Kι.
Since the graph G is (k + 2)-connected, we have at least k + 2 edges joining Kι
with G(A). In this case, at least two edges from the path S : sisi+1 and sjsj+1 are
joining Kι with G(A) , so at least two independent edges uv , u′v′ , u, u′ ∈ V(Kι) , v ,
v′ ∈ V(G(A)) , not incident with S joining Kι with G(A).
Consider the following path:
P : vus1 . . . skP (sk+1, u
′)v′ ,
where P (sk+1, u′) ⊂ Kι is a path joining sk+1 with u′ containing all vertices of
Kι \ {V(S) ∪ {u}}. We can find an example of the path P on Figure 5.
The graph G(A) is complete, so we can extend P to a cycle C containing all
vertices of G(A) and satisfying (3.3 — 3.5).
Note that as in Case 1 the path S may pass through some components Ki creating
the kind of ears of the complete graph G(A). Using the same argument as in the
54
Kι
G(A)
v
u s1
s2 sk
sk+1
v′
u′
P (sk+1, u
′)
edge from the path S.
Fig. 5: Path P containing S with both end vertices in G(A).
Subcase 1.1 we can extend the cycle C through components Ki incident with ears
preserving the properties (3.3 — 3.5).
Case 3: End vertices of S are in different components of GA i.e. s1 ∈ Kι1 ,
sk+1 ∈ Kι2 , Kι1 , Kι2 ∈ {Kι}ι∈I are such that ι1 6= ι2.
Again, since graph G is (k + 2)-connected we have at least k + 2 edges joining
every component Ki with G(A). In this case, for i = ι1 and i = ι2 at least one edge
from the path S is joining Ki with G(A) , so we have at least two independent edges
uv , u ∈ V(Kι1) , v ∈ V(G(A)) , u′v′ , u′ ∈ V(Kι2) , v′ ∈ V(G(A)) , not incident with
S joining respectively Kι1 and Kι2 with G(A).
Consider the following path:
P : vP1(u, s1)s2 . . . skP2(sk+1, u
′)v′ ,
where P1(u, s1) ⊂ Kι1 is a path joining u with s1 containing all vertices of
Kι1 \ {V(S) ∪ {u}} and P2(sk+1, u′) ⊂ Kι2 is a path joining sk+1 with u′ containing
all vertices of Kι2 \ {V(S) ∪ {u′}}. See Figure 6.
The graph G(A) is complete so we can extend P to a cycle C containing all vertices
of G(A) and satisfying (3.3 — 3.5).
Note that as in Case 1 path S may pass through several components Ki creating
the kind of ears of the complete graph G(A). Using the same argument as in Subcase
55
Kι2Kι1
G(A)
v
u s1
s2 sk
sk+1
v′
u′
P1(u, s1) P2(sk+1, u
′)
edge from the path S.
Fig. 6: Path P containing S with both end vertices in G(A).
1.1, we can extend cycle C through components Ki incident with ears preserving the
properties (3.3 — 3.5).
Case 4: One end vertex of s is in G(A) and the other end vertex is in a
component Kι of GA. In this case, we can assume without loss of generality,
that s1 ∈ G(A) and sk+1 ∈ Kι.
Since graph G is (k + 2)-connected, we have at least k + 2 edges joining the
component Kι with G(A). In this case, at least one edge from the path S is joining Kι
with G(A) , so we have at least one edge uv , u ∈ V(Kι) , v ∈ V(G(A)) , not incident
with S joining Kι with G(A).
Consider the following path:
P : s1s2 . . . skP (sk+1, u)v ,
where P (sk+1, u) ⊂ Kι0 is a path joining sk+1 with u containing all vertices of
Kι \ {V(S) ∪ {u}} , see Figure 7.
Both v and s1 are in G(A) and the graph G(A) is complete, so we can extend P
to a cycle C containing all vertices of G(A) and satisfying (3.3 — 3.5).
Note that as in Case 1, path S may pass through several components Ki creating
the kind of ears of the complete graph G(A). Using the same argument as in Subcase
56
Kι
G(A)
s1 sk
sk+1
v
u
P (sk+1, u)
edge from the path S.
Fig. 7: Path P containing S with both end vertices in G(A).
1.1, we can extend the cycle C through components Ki incident with ears preserving
the properties (3.3 — 3.5).
In all cases we have defined a cycle C containing S and now we shall extend this
cycle to a hamiltonian cycle.
Extending the cycle C to a hamiltonian cycle
We have already a cycle C satisfying conditions (3.3 — 3.5) and containing the
path S, all vertices of G(A), all vertices from components Kι containing vertices of
the path S.
Consider component Kι not included in cycle C. This component does not contain
any edge from S and since the graph G is (k+2)-connected we have at least k+2 edges
joining Kι with G(A), so at least one of these edges say uci , u ∈ V(Kι) , ci ∈ V(G(A)) ,
is not incident with S and at least one edge say u′cj , u′ ∈ V(Kι) , cj ∈ V(G(A))\ , not
incident with internal vertices of S , joining Kι with G(A). In the worst case cj = s1
or cj = sk+1.
Using these two edges uci and u′cj , we can extend cycle C through the remaining
vertices of Kι.
We consider the case cj = s1 and without loss of generality we can, assume that
on cycle C, the vertices are ordered in the following way:
s1 . . . sk+1ck+2 . . . ci . . . s1 .
57
s−1
u′
sk sk+1 c−i c
+
i
ci
u
Kι
C
s1 s2
P (u, u′)
edge from the path S.
Fig. 8: Extension of cycle C through a component Kι not incident with S.
Note that since uci is not incident with S we have c−i ci , cic
+
i 6∈ E(S) and we can
replace cycle C with the following cycle C ′
C ′ : u′s1 . . . sk+1s+k+1 . . . c
−
i s
−
1 . . . c
+
i ciP (u, u
′) ,
where P (u, u′) ⊂ Kι is a path joining u with u′ containing all vertices of Kι, see
Figure 8.
Note that this cycle C ′ satisfies conditions (3.3 — 3.5), V(C) ⊂ V(C ′) and
E(S) ⊂ E(C ′).
The case cj = sk+1 is similar.
Applying this argument for all other components Kι we can extend C to a hamil-
tonian cycle containing the path S and the proof is complete.

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[3] J.A. Bondy and U.S.R. Murty, Graph theory with applications, The Macmillan
Press LTD London 1976.
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[4] G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory Ser. B
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[5] G. Gancarzewicz and A.P. Wojda, Graphs with every k-matching in a hamiltonian
cycle, Discrete Math. 213, 2000, 141 — 151.
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J.A. Bondy and U.S.R. Murty, Academic Press N.Y. 1979, 219—231.
[7] H.V. Kronk, Variations of a theorem of Pósa in The Many Facets of Graph
Theory, ed. G. Chartrand and S.F. Kapoor, Lect. Notes Math. 110, Springer
Verlag 1969, 193—197.
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GRAPHS WITH EVERY PATH OF LENGTH k IN A HAMILTONIAN CYCLE

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GRAPHS WITH EVERY PATH OF LENGTH k IN A HAMILTONIAN CYCLE

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