The Hamiltonian problem and t -path traceable graphs
aa r X i v : . [ m a t h . C O ] A ug The Hamiltonian problem and t -path traceable graphs Kashif Bari
Department of Mathematics and StatisticsSan Diego State UniversitySan Diego, California [email protected]
Michael E. O’Sullivan
Department of Mathematics and StatisticsSan Diego State UniversitySan Diego, California [email protected]
Abstract
The problem of characterizing maximal non-Hamiltonian graphs may be naturally ex-tended to characterizing graphs that are maximal with respect to non-traceability and beyondthat to t -path traceability. We show how traceability behaves with respect to disjoint unionof graphs and the join with a complete graph. Our main result is a decomposition theoremthat reduces the problem of characterizing maximal t -path traceable graphs to characterizingthose that have no universal vertex. We generalize a construction of maximal non-traceablegraphs by Zelinka to t -path traceable graphs. The motivating problem for this article is the characterization of maximal non-Hamiltonian(MNH) graphs. Skupien and co-authors give the first broad family of MNH graphs in [6] anddescribe all MNH graphs with 10 or fewer vertices in [2]. The latter paper also includes threeconstructions—types A A A K , the Skupien MNH graphsfrom [6]. Zelinka’s second family is a broad generalization of the type A A
2, and A G , ˇ µ ( G ) and µ ( G ). The µ -invariant, introduced by Ore [5], is the maximal number of paths in G required to cover thevertex set of G . We show that ˇ µ ( G ) = µ ( G ) unless G is Hamiltonian, when ˇ µ ( G ) = 0. Maximalnon-Hamiltonian graphs are maximal with respect to ˇ µ ( G ) = 1, and maximal non-traceablegraphs are maximal with respect to ˇ µ ( G ) = 2. It is useful to broaden the perspective to study,for arbitrary t , graphs that are maximal with respect to ˇ µ ( G ) = t , which we call t -path traceablegraphs.In Section 2 we show how the ˇ µ and µ invariants behave with respect to disjoint union ofgraphs and the join with a complete graph. Section 3 derives the main result, a decompositiontheorem that reduces the problem of characterizing maximal t -path traceable to characterizingthose that have no universal vertex, which we call trim. Section 4 presents a generalization ofthe Zelinka construction to t -path traceable graphs.1 Traceability and Hamiltonicity
It will be notationally convenient to say that the complete graphs K and K are Hamiltonian. Asjustification for this view, consider an undirected graph as a directed graph with each edge havinga conjugate edge in the reverse direction. This perspective does not affect the Hamiltonicity ofa graph with more than 3 vertices, but it does give K a Hamiltonian cycle. Similarly, addingloops to any graph with more than 2 vertices does not alter the Hamiltonicity of the graph, but K , with an added loop, has a Hamiltonian cycle.Let G be a graph. A vertex, v ∈ V ( G ) , is called a universal vertex if deg( v ) = | V ( G ) | − G denote the graph complement of G , having vertex set V ( G ) and edge set E ( K n ) \ E ( G ).We will use the disjoint union of two graphs, G ⊔ H and the join of two graphs G ∗ H . Thelatter is G ⊔ H together with the edges { vw | v ∈ V ( G ) and w ∈ V ( H ) } . Definition 1.
A set of s disjoint paths in a graph G that includes every vertex in G is a s -pathcovering of G . Define the following invariants. µ ( G ) := min s ∈ N {∃ s -path covering of G } .ˇ µ ( G ) := min l ∈ N { K l ∗ G is Hamiltonian } i H ( G ) := ( G is Hamiltonian0 otherwiseWe will say G is t -path traceable when µ ( G ) = t . A set of t disjoint paths that cover a t -pathtraceable graph G is a minimal path covering .Note that K r ∗ ( K s ∗ G ) = K r + s ∗ G . If G is Hamiltonian then so is K r ∗ G for r >
0. (Inparticular this is true for G = K and G = K .)We now have a series of lemmas that lead to the main result of this section, which is aformula showing how the µ -invariant and ˇ µ -invariant behave with respect to disjoint union andthe join with a complete graph. Lemma 2. ˇ µ ( G ) = min l ∈ N { K l ∗ G is Hamiltonian } Proof.
Since K l ∗ G is a subgraph of K l ∗ G , a Hamiltonian cycle in K l ∗ G would also be one in K l ∗ G .Let ˇ µ ( G ) = a . Suppose C is a Hamiltonian cycle in K a ∗ G and write C as v ∼ P ∼ Q ∼ . . . ∼ P s ∼ Q s ∼ v , where v is a vertex in G and the paths P i ∈ G and Q i ∈ K a . If any Q i contains 2 vertices or more, say u and w , . . . , w k with k >
1, then we may simply removeall the vertices, except u , and end up with a Hamiltonian graph on K a − k . This contradictsthe minimality of a = ˇ µ ( G ). Therefore, C must not contain any paths of length greater thantwo in the subgraph K a , and any Hamiltonian cycle on K a ∗ G is also a Hamiltonian cycle on K a ∗ G . Lemma 3. ˇ µ ( G ) = µ ( G ) − i H ( G ) Proof. If G is Hamiltonian (including P and P ) then ˇ µ ( G ) = 0, µ ( G ) = 1 so the equalityholds. Suppose G is non-Hamiltonian with µ ( G ) = t and t -path covering P , . . . , P t . Let K t u , . . . , u t . In the graph K t ∗ G , there is a Hamiltonian cycle: v ∼ P ∼ v ∼ P ∼· · · ∼ v t ∼ P t ∼ v . Thus ˇ µ ( G ) t = µ ( G ).Let ˇ µ ( G ) = a , so there is a Hamiltonian cycle in K a ∗ G . Removing the vertices of K a breaksthe cycle into at most a disjoint paths covering G . Thus µ ( G ) ˇ µ ( G ). Lemma 4. µ ( G ⊔ H ) = µ ( G ) + µ ( H ) and ˇ µ ( G ⊔ H ) = ˇ µ ( G ) + ˇ µ ( H ) + i H ( G ) + i H ( H ) .Proof. A path covering of G may be combined with a path covering of H to create one for G ⊔ H .Conversely, paths in a t -path covering of G ⊔ H can be partitioned into those contained in G and those contained in H , giving a path covering of G and one of H . Consequently µ ( G ⊔ H ) = µ ( G ) + µ ( H )Since G ⊔ H is not Hamiltonian we haveˇ µ ( G ⊔ H ) = µ ( G ⊔ H ) + i H ( G ⊔ H )= µ ( G ) + µ ( H )= ˇ µ ( G ) + i H ( G ) + ˇ µ ( H ) + i H ( H ) Lemma 5.
For any graph G , µ ( K s ∗ G ) = max { , µ ( G ) − s } ˇ µ ( K s ∗ G ) = max { , ˇ µ ( G ) − s } In particular, if K s ∗ G is Hamiltonian then µ ( K s ∗ G ) = 1 and ˇ µ ( K s ∗ G ) = 0 ; otherwise, µ ( K s ∗ G ) = µ ( G ) − s and ˇ µ ( K s ∗ G ) = ˇ µ ( G ) − s .Proof. The formula for ˇ µ is immediate when G is Hamiltonian since we have observed that thisforces K s ∗ G to be Hamiltonian. Otherwise, it follows from K r ∗ ( K s ∗ G ) = K r + s ∗ G : ifˇ µ ( G ) = a , then K r ∗ ( K s ∗ G ) is Hamiltonian if and only if r + s > a .The formula for µ may be derived from the result for ˇ µ using Lemma 3. We may also proveit directly. Observe that it is enough to prove µ ( K ∗ G ) = max { , µ ( G ) − } . Let u be thevertex of K . Let µ ( G ) = t and P , . . . , P t a t -path covering of G . If t = 1 then u can beconnected to the initial vertex of P to create a 1-path covering of K ∗ G . For t >
2, the path P ∼ u ∼ P along with P , . . . , P t gives a ( t − K ∗ G . Thus for t > µ ( K ∗ G ) t −
1. Suppose Q , . . . , Q d were a minimal d -path covering of K ∗ G , with u avertex of Q . Removing u gives at most a ( d + 1)-path covering of G . Thus µ ( K ∗ G ) + 1 > t .This shows µ ( K ∗ G ) = µ ( G ) − µ ( G ) > µ and ˇ µ invariantsfor the disjoint union of graphs, and the join with a complete graph.3 roposition 6. Let { G j } mj =1 be graphs. µ (cid:0) m G j =1 G j (cid:1) = m X j =1 µ ( G j ) and ˇ µ (cid:0) m G j =1 G j (cid:1) = m X j =1 ˇ µ ( G j ) + m X j =1 i H ( G j ) .Furthermore, ˇ µ (cid:0) ( m G j =1 G j ) ∗ K r (cid:1) = max (cid:8) , m X j =1 ˇ µ ( G j ) + m X j =1 i H ( G j ) − r (cid:9) .Proof. We proceed by induction. The base case k = 2 is exactly Lemma 4. Assume the formulaholds for k graphs we will prove it for k + 1 graphs. µ (cid:0) k +1 G j =1 G j (cid:1) = µ (cid:0) ( k +1 G j =1 G j ) ⊔ G k +1 (cid:1) = µ (cid:0) k G j =1 G j (cid:1) + µ (cid:0) G k +1 (cid:1) = k X j =1 µ ( G j ) + µ (cid:0) G k +1 (cid:1) = k +1 X j =1 µ ( G j )By Lemma 3 and the fact that disjoint graphs are not Hamiltonian, we have,ˇ µ (cid:0) m G j =1 G j (cid:1) = µ (cid:0) m G j =1 G j (cid:1) + i H (cid:0) m G j =1 G j (cid:1) = m X j =1 µ ( G j ) + 0= m X j =1 (ˇ µ ( G j ) + i H ( G j ))= m X j =1 ˇ µ ( G j ) + m X j =1 i H ( G j )Therefore, we have by Lemma 5,ˇ µ (cid:0) ( m G j =1 G j ) ∗ K r (cid:1) = max { , ˇ µ (cid:0) m G j =1 G j (cid:1) − r } = max { , m X j =1 ˇ µ ( G j ) + m X j =1 i H ( G j ) − r } G and vertex v ∈ G , T ( v, G ) is true if and only if v is a terminal vertex in some minimal path covering of G . Lemma 7.
Let v ∈ G and w ∈ H . µ (cid:16) ( G ⊔ H ) + vw (cid:17) = ( µ ( G ⊔ H ) − if T ( v, G ) and T ( w, H ) µ ( G ⊔ H ) otherwiseProof. Let µ ( G ) = c , µ ( H ) = d and µ (cid:16) ( G ⊔ H ) + vw (cid:17) = t . Clearly, t c + d .Let R , . . . , R t be a minimal path cover of ( G ⊔ H ) + vw . If no R i contains vw then this isalso a minimal path cover of ( G ⊔ H ) so t = c + d . Suppose R contains vw and note that R isthe only path with vertices in both G and H . Removing vw gives two paths P ⊆ G and Q ⊆ H .Paths P and Q along with R , . . . , R t cover G ⊔ H , so t + 1 > c + d . Thus, t can either be c + d or c + d − t = c + d −
1, then we have the minimal ( t + 1)-path covering P, Q, R , . . . , R t of G ⊔ H ,as above. We note that v must be a terminal point of P and w must be a terminal point of Q ,by construction. This path covering may be partitioned into a c -path covering of G containing P and a d -path covering of H containing Q . Thus, T ( v, G ) and T ( w, G ) hold.Conversely, suppose T ( u, G ) and T ( w, H ) both hold. Let P , . . . , P c be a minimal path of G with v a terminal vertex of P and let Q , . . . , Q d be a minimal path cover of H with w a terminalvertex of Q . The edge vw knits P and Q into a single path and P ∼ Q , P , . . . , P c , Q , . . . , Q d is a c + d − G ⊔ H ) + vw . Consequently, t c + d − T ( u, G ) and T ( w, H ) both hold if and only if t = c + d −
1. Otherwise, t = c + d . Corollary 8.
Let v ∈ G and w ∈ H . ˇ µ (cid:16) ( G ⊔ H ) + vw (cid:17) = ˇ µ ( G ⊔ H ) − if G = H = K ˇ µ ( G ⊔ H ) − if T ( v, G ) and T ( w, H )ˇ µ ( G ⊔ H ) OtherwiseProof.
Let δ = 1 if T ( v, G ) and T ( w, H ) are both true and δ = 0 otherwise. Thenˇ µ (cid:16) ( G ⊔ H ) + vw (cid:17) = µ (cid:16) ( G ⊔ H ) + vw (cid:17) − i H (cid:16) ( G ⊔ H ) + vw (cid:17) = µ (( G ⊔ H ) − δ − i H (cid:16) ( G ⊔ H ) + vw (cid:17) The final term is − G = H = K . t -path traceable graphs In this section we prove our main result, a maximal t -path traceable graph may be uniquelywritten as the join of a complete graph and a disjoint union of graphs that are also maximal5ith respect to traceability, but which are also either complete or have no universal vertex. Wework with the families of graphs M t for t > N t for t > M t := { G | ˇ µ ( G ) = t and ˇ µ ( G + e ) < t, ∀ e ∈ E ( G ) } N t := { G ∈ M t | G is connected and has no universal vertex } The set M is the set of complete graphs. The set M is the set of graphs with a Hamiltonianpath but no Hamiltonian cycle, that is, maximal non-Hamiltonian graphs. For t > M t is alsothe set of graphs G such µ ( G ) = t and µ ( G + e ) = t − e ∈ E ( G ). We will call these maximal t -path traceable graphs . A graph in N t will be called trim . Proposition 9.
For s < t , G ∈ M t if and only if K s ∗ G ∈ M t − s .Proof. We have ˇ µ ( K s ∗ G ) = ˇ µ ( G ) − s , so we just need to show that K s ∗ G is maximal if andonly if G is maximal. The only edges that can be added to K s ∗ G are those between vertices of G , that is, E ( K s ∗ G ) = E ( G ). For such an edge e ,ˇ µ (cid:16) ( K s ∗ G ) + e (cid:17) = ˇ µ (cid:16) K s ∗ ( G + e ) (cid:17) = ˇ µ ( G + e ) − s (1)Consequently, ˇ µ ( G + e ) = ˇ µ ( G ) − µ (cid:16) ( K s ∗ G ) + e (cid:17) = ˇ µ ( K s ∗ G ) − s = t > K s ∗ G will not be a complete graphand M is the set of complete graphs. The proof breaks down in (1). Proposition 10.
Let G ∈ M c and H ∈ M d . The following are equivalent.1. G ⊔ H ∈ M c + d + i H ( G )+ i H ( H )
2. Each of G and H is either complete or has no universal vertex.Proof. We have already shown that ˇ µ ( G ⊔ H ) = c + d + i H ( G ) + i H ( H ). We have to considerwhether adding an edge to G ⊔ H reduces the ˇ µ -invariant. There are three cases to consider,the extra edge may be in E ( G ) or E ( H ) or it may join a vertex in G to one in H . Since G is maximal, adding an edge to G is either impossible, when G is complete, or it reduces theˇ µ -invariant of G . This edge would also reduce the ˇ µ -invariant of G ⊔ H by Lemma 4. The casefor adding an edge of H is the same. Consider the edge vw for v ∈ V ( G ) and w ∈ V ( H ). ByCorollary 8 the ˇ µ -invariant will drop if and only if v is the terminal point of a path in a minimalpath covering of G and similarly for w in H , that is, T ( v, G ) and T ( w, H ). Clearly this holdsfor all vertices in a complete graph. The following lemma shows that T ( v, G ) holds for G ∈ M c with c > v is not a universal vertex in G . Thus, in order for G ⊔ H to be maximal G must either be complete, or be maximal itself, and have no universal vertex, and similarly for H . As a key step before the main theorem, the next lemma shows that in a maximal graph,each vertex is universal, or a terminal vertex in a minimal path covering.6 emma 11. Let c > and G ∈ M c . For any two non-adjacent vertices v, w in G there is a c -path covering of G in which both v and w are terminal points of paths. Moreover, a vertex v ∈ G is a terminal point in some c -path covering if and only if v is not universal.Proof. Suppose c > v, w be non-adjacent in G . Since G is maximal G + vw has a( c − P , . . . , P c − . The edge vw must be contained in some P i because G has no( c − c -path covering of G with v and w as terminalvertices. The special case c = 1 is well known, adding the edge vw gives a Hamiltonian cycle,and removing it leaves a path with endpoints v and w . A consequence is that any non-universalvertex is the terminal point of some path in a c -path covering.Suppose P , . . . , P c is a c -path covering of G ∈ M c with v a terminal point of P i . Then v is not adjacent to any of the terminal points of P j for j = i , for otherwise two paths could becombined into a single one. In the case c = 1, v cannot be adjacent to the other terminal pointof P , otherwise G would have a Hamiltonian cycle. Consequently a universal vertex is not aterminal point in a c -path covering of G . Theorem 12.
For any G ∈ M t , t > , G may be uniquely decomposed as K s ∗ ( G ⊔ . . . ⊔ G r ) ,where s is the number of universal vertices of G , and each G j is either complete or G j ∈ N t j for some t j > . Furthermore t = r X j =1 t j + r X j =1 i H ( G j ) − s .Proof. Suppose G ∈ M t and let s be the number of universal vertices of G . Let r be the numberof components in the graph obtained by removing the universal vertices from G , let G , . . . G r be the components and let ˇ µ ( G j ) = t j .Proposition 6 shows that t = r X j =1 t j + r X j =1 i H ( G j ) − r . By Proposition 9, we have that G ∈ M t if and only if G ⊔ . . . ⊔ G r ∈ M t + s . Furthermore, each G j must be in M t j for otherwise wecould . Without loss of generality if we add an edge e to G , such that ˇ µ ( G + e ) < t , thenˇ µ ( G + e ) = ˇ µ ( G + e ) + r X j =2 t j + r X j =1 i H ( G j ) − s< r X j =1 t j + r X j =1 i H ( G j ) − s = t Now, we apply Proposition 10, so then G ⊔ . . . ⊔ G r ∈ M t + s , where t + s = r X j =1 t j + r X j =1 i H ( G j )if and only if G j is either trim or complete. In other words, G j ∈ N t j for t j > G j ∈ M for t j = 0. 7 v v u u u Figure 1: Smallest graph in N t -path traceable graphs Skupien [6] discovered the first family of maximal non-Hamiltonian graphs, that is, graphs in M . These graphs are formed by taking the join with K r of the disjoint union of r + 1 completegraphs. The smallest graph in N is shown in Figure 1. Chv´atal identified its join with K as thesmallest maximal non-Hamilitonian graph that is not 1-tough, that is, not one of the Skupienfamily. Jamrozik, Kalinowski and Skupien [2] generalized this example to three different families.Family A u i v i with an arbitrary complete graph containing u i and replacesthe K formed by the u i with an arbitrary complete graph. The result has four cliques, the firstthree disjoint from each other but each intersecting the fourth clique in a single vertex. Thisgraph is also in N and its join with K gives a maximal non-Hamiltonian graph. Family A K of the disjoint union of a complete graph and the graphin N just described. Theorem 12 shows that the resulting graph is in M . Family A A N . Bullock, Frick,Singleton and van Aardt [1] recognized that two constructions of Zelinka [7] gave maximal non-traceable graphs, that is, elements of M . Zelinka’s first construction is like the Skupien family:formed from r + 1 complete graphs followed by the join with K r − . The Zelinka Type II familycontains graphs in N that are a significant generalization of the graphs in Figures 1 and 2. Inthis section we generalize this family further to get graphs in N t for arbitrary t . Our startingpoint is the graph in Figure 3, which is in N . Example 13.
Consider K m with m > t − u , . . . , u m . Let G be the graphcontaining K m along with vertices v , . . . , v t − and edges u i v i . The case with t = 3 and m = 5 = 2 t − G ∈ N t .One can readily check that this graph is t -path covered using v i − ∼ u i − ∼ u i ∼ v i for i = 1 , . . . , t − v t − ∼ u t − ∼ u t ∼ · · · ∼ u m . We check that G is maximal. By thesymmetry of the graph, we need only consider the addition of the edge v u m and v u . In eithercase, the last and the first paths listed above may be combined into one, either v t − ∼ u t − ∼ · · · ∼ u m ∼ v ∼ u ∼ u ∼ v , or v t − ∼ u t − ∼ · · · ∼ u m ∼ u ∼ v ∼ u ∼ v u v u u u v v Figure 2: The join of this graph with K is the smallest graph in the A t − t -path covered graph with 2 t − Lemma 14.
Let G be a connected graph and let u , v , v , v ∈ G with deg( v i ) = 1 , and u adjacent to v and v but not v . Then µ ( G ) = µ ( G + uv ) .Proof. Let P , . . . , P r be a minimal path covering of G + uv ; it is enough to show that thereare r -paths covering G . If the covering doesn’t include uv , then P , . . . , P r also give a minimalpath covering of G establishing the claim of the lemma. Otherwise, suppose uv is an edge of P . We consider two cases.Suppose P contains the edge uv (or similarly uv ). Then P has v as a terminal point andone of the other paths, say P must be a length-0 path containing simply v . Let Q be obtainedby removing uv and uv from P . Then v ∼ u ∼ v , Q, P , . . . , P r , gives an r -path covering of G Suppose P contains neither uv nor uv . Then each of v and v must be on a length-0 pathin the covering, say P and P are these paths. Furthermore u must not be a terminal point of P ,for, if were, the path could be extended to include v or v , reducing the number of paths requiredto cover G . Removing u from P yields two paths, Q , Q . Then v ∼ u ∼ v , Q , Q , P , . . . , P r gives an r -path cover of G . This proves the lemma. Proposition 15.
Let G ∈ N t . The number of degree-one vertices in G is at most t − . Thisoccurs if and only if the t − vertices of degree-one have distinct neighbors and removing thedegree-one vertices leaves a complete graph.Proof. Each degree-one vertex must be a terminal point in a path covering. So any graph G covered by t paths can have at most 2 t degree-one vertices. Aside from the case t = 1 and9 u u u u v v v v v Figure 3: Whirligig in N . G = K , we can see that a graph with 2 t degree-one vertices cannot be maximal t -path traceableas follows. It is easy to check that a 2 t star is not t -path traceable (it is also not trim). A t -pathtraceable graph with 2 t degree-one vertices must therefore have an interior vertex w that is notconnected to one of the degree-one vertices v . Such a graph is not maximal because the edge vw can be added leaving 2 t − t − G ∈ N t with 2 t − v , . . . , v t − . Lemma 14 shows thatno two of the v i can be adjacent to the same vertex, for that would violate maximality of G .So, the v i have distinct neighbors. Furthermore, all the nodes except the v i can be connectedto each other and a path covering will still require at least t paths since there remain 2 t − N t .We can now generalize the Zelinka family. Construction 16.
Let U , U , · · · U t − be disjoint sets and U = t − G i =0 U i . Let m i = | U i | andassume that for i > U i are non-empty, so m i >
0. For i = 1 , . . . , t − i = 0)and j = 1 , . . . , m i , let V ij be disjoint from each other and from U . Form the graph with vertexset U ⊔ (cid:16) t − G i =1 (cid:0) m i G j =1 V ij (cid:1)(cid:17) and edges uu ′ for u, u ′ ∈ U and uv for any u ∈ U i and v ∈ V ij with i = 1 , . . . , t − j = 1 , . . . , m i . The cliques of this graph are K U and K U i ⊔ V ij for each i = 1 , . . . , t − j = 1 , . . . , m i .The graph in Figure 2 has m = 0, m = m = 1 and m = 2, and the graph in Figure 4indicates the general construction. 10 U U . . . U t − U t − U t − V , . . . V ,m V t − , . . . V t − ,m t − V , . . . V ,m V t − , . . . V t − ,m t − V t − , ...V t − ,m t − Figure 4: Generalization of the Whirligig, W Theorem 17.
The graph W in Construction 16 is a trim, maximal t -path traceable graph.Proof. We must show that W is t -path covered and not ( t − t − R be a Hamiltonian path in U . For each i = 1 , . . . , t − j = 1 , . . . , m i let Q ij bea Hamiltonian path in K V ij . Let P i be the path P i : Q i ∼ u i ∼ · · · ∼ Q im i ∼ u im i and let ←− P i be the reversal of P i .Since there is an edge u im i u jm j there is a path P i ∼ ←− P j for any i = j ∈ { , . . . , t − } .Therefore the graph W has a t -path covering P i − ∼ ←− P i for i = 1 , . . . , ( t −
1) , along with P t − ∼ R . We leave to the reader the argument that there is no ( t − W is maximal we show that after adding an edge e , we can join two paths in the t -path cover above, with a bit of rearrangement. There are three types of edges to consider, theedge e might join V ij to U i ′ for i = i ′ ; or V ij to V ij ′ for j = j ′ ; or V ij to V i ′ j ′ for i = i ′ . Becauseof the symmetry of W , we may assume i = 1 and j = 1 and that the vertex chosen from V ij isthe initial vertex of Q ij . Other simplifications due to symmetry will be evident in what follows.In the first case there are two subcases—determined by i ′ > t or not—and after permutation,we may consider the edge e from the initial vertex of Q to the terminal vertex of R , or to theterminal vertex of P t − . We can then join two paths in the t -path cover: either P t − ∼ R e ∼ P ∼ ←− P or P ∼ ←− P e ∼ P t − ∼ R . 11uppose next that we join the initial vertex of Q with the terminal vertex of Q . We thenrearrange P and join two path in the t -path cover to get P t − ∼ R ∼ u ∼ Q e ∼ Q ∼ u ∼ · · · ∼ Q m ∼ u m ∼ ←− P Finally, suppose that we join the initial vertex of Q with the initial vertex of Q t − , . Thenwe rearrange to ←− R ∼ ←− P t − e ∼ P ∼ ←− P . References [1] F. Bullock, M. Frick, J. Singleton, S. van Aardt, K. Mynhardt,
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Discrete Mathematics (1991), 213-220.[4] S. Noorvash, Covering the vertices of a graph by vertex-disjoint paths,
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Ann. Mat. Ser. IV (1961), 315-321.[6] Z. Skupien, On Maximum non-Hamiltonian graphs , Rostock. Math. Kolloq. (1979),97-106.[7] B. Zelinka, Graphs maximal with respect to absence of Hamiltonian Paths,
Discus-siones Mathematicae, Graph Theory18