Hamilton-Connected Mycielski Graphs

All graphs considered in this paper are simple and finite. For notations and terminologies not defined here, we refer to Bondy and Murty [1]. A spanning cycle (path) of a graph is called Hamilton cycle (Hamilton path). A graph which contains a Hamiltonian path between every two vertices of G is called Hamilton-connected (HC). Mycielski [2] proved that the chromatic numbers of triangle-free graphs can be arbitrarily large by introducing a graph transformation as follows. For a graph G on vertices V � v1, v2, . . . , vn 􏼈 􏼉, its Mycielski graph, denoted by μ(G), is the graph on vertices X∪Y∪ z { } � x1, x2, . . . , xn 􏼈 􏼉∪ y1, y2, . . . , yn 􏼈 􏼉∪ z { } with edges zyi for all i and edges xixj, yixj, and xiyj for all edges vivj in G. In recent years, a number of papers are devoted to various properties of Mycielski graphs, such as Hamiltonconnectivity, Hamiltonicity [3–7], total chromatic number [8, 9], circular chromatic number [10–14], and connectivity [15, 16]. Fisher et al. [4] obtained the following results.


Introduction
All graphs considered in this paper are simple and finite. For notations and terminologies not defined here, we refer to Bondy and Murty [1]. A spanning cycle (path) of a graph is called Hamilton cycle (Hamilton path). A graph which contains a Hamiltonian path between every two vertices of G is called Hamilton-connected (HC). Mycielski [2] proved that the chromatic numbers of triangle-free graphs can be arbitrarily large by introducing a graph transformation as follows. For a graph G on vertices V � v 1 , v 2 , . . . , v n , its Mycielski graph, denoted by μ(G), is the graph on vertices X ∪ Y ∪ z { } � x 1 , x 2 , . . . , x n ∪ y 1 , y 2 , . . . , y n ∪ z { } with edges zy i for all i and edges x i x j , y i x j , and x i y j for all edges v i v j in G. In recent years, a number of papers are devoted to various properties of Mycielski graphs, such as Hamiltonconnectivity, Hamiltonicity [3][4][5][6][7], total chromatic number [8,9], circular chromatic number [10][11][12][13][14], and connectivity [15,16]. Fisher et al. [4] obtained the following results.
e following results hold for a graph G:

(1) If G is Hamiltonian, then μ(G) is Hamiltonian (2) If G is not connected, then μ(G) is not Hamiltonian (3) If G has at least two pendant vertices, then μ(G) is not Hamiltonian
Cheng, Wang, and Liu studied Hamiltonicity and Hamilton-connectedness in Mycielski graphs of bipartite graphs.
Theorem 2 (see Cheng et al. [3]). For a bipartite graph G, the following are true:

is Hamiltonian, then G has a Hamilton path
In 2017, Jarnicki et al. [17] established the following results for μ(G) being Hamilton-connected or not. Theorem 3 (see Jarnicki et al. [17]). e following results hold for a graph G: (1) If G is an odd cycle, then μ(G) is Hamilton-connected (2) If G is a Hamilton-connected graph with order odd, then μ(G) is Hamilton-connected (3) If G is an even cycle, then μ(G) is not Hamiltonconnected ey posed the following conjecture.
Conjecture 1 (see Jarnicki et al. [17]). If G is Hamiltonconnected and not K 2 , then μ(G) is Hamilton-connected. In this paper, we confirm that the conjecture is true for three families of graphs: the graphs G with δ(G) > |V(G)|/2, generalized Petersen graphs GP(n, 2) and GP(n, 3), and the cubes G 3 . In addition, if G is pancyclic, then μ(G) is pancyclic.

Mycielski Factor
Let G be a connected graph of order n even, and v 1 ∈ V(G). We call a connected spanning subgraph of G to be a Mycielski factor starting at v 1 if it consists of an even number of odd cycles C 1 , . . . , C 2s (possibly s � 0) and an even cycle C 2s+1 with the chord (possibly empty), joined by 2s edges , and the chord joins v 2s+1 ′ and a vertex at distance even on C 2s+1 . Lemma 1. Assume that a graph G is Hamilton-connected. If, for any v ∈ V(G), there exists a Mycielski factor starting at v, then μ(G) is Hamilton-connected.
Proof. As in the assumption, let G be HC. Trivially, G has a Hamilton cycle. By eorem 3 (2), μ(G) is HC if the order of G is odd. So, it remains to tackle the case when the order is

Take any two vertices
A, B ∈ V(μ(G)). We consider five cases in terms of the location of A and B in X, Y, and z { }.
□ Case 1. A ∈ X and B ∈ X. Without loss of generality, let A � x 1 and B � x 2n . Since G is HC, there exists a Hamilton path P connecting v 1 and v 2n in G. We shall find a Hamilton path of μ(G) depending on P as follows. Zigzag up from x 1 until y 2n is reached. en, jump via z to y 1 , and zigzag right until x 2n is reached. Formally, it is as shown in Figure 1.
Without loss of generality, let A � y 1 and B � y 2n . Since G is HC, there exists a Hamilton path P connecting v 1 and v 2n in G. us, there exists a neighbor, say v 2 , of v 1 . Zigzag up from y 1 to x 2 and then back to x 1 , zigzag up to x 2n− 1 and then up to x 2n , and zigzag left to y 3 and then up to z and y 2n , as shown in Figure 2. Formally, Without loss of generality, let A � x 1 and B � y 2n . Since G is HC, there exists a Hamilton path P connecting v 1 and v 2n in G. We are able to find a Hamilton path joining A and B: zigzag up from x 1 to x 2n− 1 and then up to x 2n , and zigzag left to y 1 , and then reach z and y 2n , as shown in Figure 3. Formally, it is Without loss of generality, let A � y 1 . Since G is HC, G has a Hamilton cycle C. Label the vertices of C as v 1 v 2 , . . . , v 2n v 1 . We are able to find a Hamilton path joining A and B: zigzag from y 1 to x 2n and then go to x 1 , and then zigzag right to y 2n , and finish at z, as shown in Figure 4. Formally, it is Case 5. A ∈ X and B � z.
Without loss of generality, let A � x 1 . Let H be a Mycielski factor of G starting at v 1 , which consists of an even number of odd cycles C 1 , . . . , C 2s (possibly s � 0) and an even cycle C 2s+1 with the chord (possibly empty), joined by 2s edges e 1 , . . . , e 2s , where , and the chord joins v 2s+1 ′ and a vertex at distance even on C 2s+1 .
{ }, label the vertices of C i in the clockwise order u 1 u 2 , . . . , u 2k+1 . One can find a Hamilton path P i of μ(C i ) as follows: where u 1 � v i and u 2k+1 � v i ′ . Let C 2s+1 � w 1 w 2 · · · w 2l be an even cycle with chord w 2l w 2t in H. One can find a Hamilton path P 2s+1 of μ(C i ) as follows: where

Hamiltonian Connectedness
Proof. Let v be a vertex of G. We consider a Hamilton cycle C of G. Let u be a neighbor of v on C. Since d(u) ≥ (n/2) + 1, it has a neighbor at distance even on C. By Lemma 1, μ(G) is HC.
e k th power of a graph G, denoted by G k , is a graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is at most k. us, We need the following result due to Karaganis [18]. □ Theorem 5 (see Karaganis [18]). e cube G 3 of every connected graph G of order n ≥ 3 is Hamilton-connected. Theorem 6. For any connected graph G of order n ≥ 3, μ(G 3 ) is Hamilton-connected.
Proof. By eorem 5, G 3 is HC for G. Since μ(H 3 ) is a spanning subgraph of μ(G 3 ) for any spanning graph H of G, to show μ(G 3 ) is HC, it suffices to show that μ(T 3 ) is HC for any tree T of order n ≥ 3. Since T 3 is HC, by eorem 3 (1), we may assume that n is even. By Lemma 1, it remains to show that T 3 has a Mycielski factor starting from each vertex v ∈ V(T).
Let w be a neighbor of v in T, and let T v and T w be the components of T − vw containing v and w, respectively. Let n v and n w be the order of T v and T w , respectively. Let v ′ be a neighbor of v in T v , and let w ′ be a neighbor of w in T w .
Case 1: both n v and n w are at least 3. Subcase 1.1: both n v and n w are odd. By eorem 5, both T 3 v and T 3 w are HC. Let C v and C w be Hamilton cycles of T v and T w , respectively. One Since n v + n w � n is an even number at least 3, n w is an odd number at least 3. By eorem 5, let C ww ′ be a Hamilton cycle of T 3 w containing ww ′ . It is easy to see that C ww ′ + vw + vw ′ is a Mycielski factor of T 3 starting at v. Subcase 2.1.2: min n v , n w � n v � 2. If n w � 2, then n � 4. Trivially, T 3 w � K 4 has a Mycielski factor starting at v. If n w ≠ 2, then n w is an even number at least 4. By eorem 5, let P ww′ be a Hamilton path of T 3 w . It can be seen that factor starting at v.
If T � K 1,n− 1 , then T 3 � K n has a Mycielski factor starting at v. Next, we assume that T ≇ K 1,n− 1 . We can choose a neighbor w of v such that n w ≥ 2. Combining with our assumption that min n v , n w � n w ≤ 2, we have n w � 2. By eorem 5, let P vv′ be a Hamilton path of T 3 v . It can be checked that P vv′ + vw + ww ′ + v ′ w ′ + v ′ w is a Mycielski factor starting at v.

Theorem 8. If
Proof. In view of Lemma 1, it suffices to show that GP(n, 2) has a Mycielski factor starting at any v ∈ V (GP(n, 2)). We consider two cases: Case 1: n ≡ 1 or 3(mod 6). Since n is odd, GP(n, 2) is vertex-transitive, and we may assume that v � u 1 , without loss of generality. Let C 1 and C 2 be the outer cycle and inner cycle of GP(n, 2). Let v be a vertex of GP(n, 2). It is clear that C 1 ∪ C 2 + u 2 v 2 is a Mycielski factor of GP(n, 2) starting at v. Case 2: n ≡ 2(mod 6).
By the symmetry, it suffices to tackle two possibilities according to the location of v in GP(n, 2): v lies on the outer cycle or inner cycle of GP(n, 2). Without loss of generality, First, for the case when n � 8, we can find a Mycielski factor F n of GP(n, 2) as follows: Discrete Dynamics in Nature and Society where For n � 14, by inserting 12 new vertices to C 8 of F 8 , we get C 14 as illustrated in Figures 5-7 for the case that v lies in the outer cycle and for the case that v lies in the inner cycle as illustrated in Figures 6, 8, and 9. For the case when n ≥ 20, by inserting 12 new vertices to F n− 6 with type A insertion, we obtain a Mycielski factor F n of GP(n, 2) starting at v. □ Theorem 9 (see Alspach and Liu [21]). e generalized Petersen graph GP(n, 3) with n ≥ 6 is Hamilton-connected if and only if n is odd. If GP(n, 3) is Hamilton-connected, then μ (GP(n, 3)) is Hamilton-connected.
Proof. Since GP(n, 3) is Hamilton-connected, by eorem 9, n is an odd number at least 7. Let v be a vertex of GP(n, 3). In view of Lemma 1, it suffices to show that GP(n, 3) has a Mycielski factor starting at v. We consider two cases: Case 1: n ≡ 1 or 5 (mod 6). Since n is odd, GP(n, 3) is vertex-transitive; by the symmetry, we may assume that v � u 1 , without loss of generality. Let C 1 and C 2 be the outer cycle and inner cycle of GP(n, 3). It is clear that C 1 ∪ C 2 + u 2 v 2 is a Mycielski factor of GP(n, 3) starting at v. Case 2: n ≡ 3(mod 6).
By the symmetry, it suffices to tackle two possibilities according to the location of v in GP(n, 3): v lies on the outer cycle or inner cycle of GP(n, 3). Without loss of generality, First, for the case when n � 9, we can find a Mycielski factor F n of GP(n, 3) starting at v as follows: where For n � 15, by inserting 12 new vertices to C 9 of F 9 , we get C 15 as illustrated in Figures 10-12. For the case when n ≥ 21, by inserting 12 new vertices to F n− 6 with type B insertion, we obtain a Mycielski factor F n of GP(n, 3) starting at v.

Pancyclicity
In this section, we show that if a graph G is pancyclic, then μ(G) is also pancyclic.

Theorem 11. If G is pancyclic, then μ(G) is pancyclic.
Proof. Let G be a pancyclic graph of order n. Since μ(G) contains G as its subgraph, μ(G) contains a cycle of length l for each l ∈ 3, . . . , n { }. Now, we find a cycle of length n + 1 in μ(G). Take a cycle C of length n − 2 in G. Without loss of generality, let P � v 1 v 2 , . . . , v n− 2 be a path resulting from C deleting an edge. It can be seen that x 1 x 2 , . . . , x n− 2 y 1 zy n− 2 x 1 is a cycle of length n + 1, as illustrated in Figure 13. In a similar way, one can find a cycle of length n + 2 in μ(G) in terms of a cycle of length n − 1 in G.
Next, we will find a cycle of length n + k in μ(G) for each k ∈ 3, . . . , n +   Discrete Dynamics in Nature and Society generality, let C � v 1 v 2 , . . . , v n v 1 in G. We consider two cases according to the parity of k: One can find a cycle of length n+k in μ(G), as shown in Figure 14. Formally, it is Case 2: k is even. Zigzag up from x 1 to x k− 1 and left to x k− 2 , then zigzag left to y 1 , z, y k− 1 , and x k , and go right to x n and back to x 1 , as shown in Figure 15. Formally, it is □ u 1 u 2 u 7 u 8 Figure 5: F 8 from u 1 to u 2 in GP (8,2).

Conclusion
In this paper, we introduce the notion of the Mycielski factor of a graph. If a graph G has a Mycielski factor starting at v for any v ∈ V(G), then μ(G) is Hamilton-connected. Applying this result, we are able to show that if a graph G belongs to three (well-defined) families of graphs, then μ(G) is Hamilton-connected. However, the full conjecture of Jarnicki, Myrvold, Saltzman, and Wagon is not yet solved. We also prove that if G is pancyclic, then μ(G) is pancyclic.
One of the reviewers proposed the following two interesting problems.
Zhong et al. [7] showed that the line graph of the generalized Petersen graph GP(n, k) is always Hamiltonconnected. Is it easy to show that the Mycielski graph of L (GP(n, k)) is Hamilton-connected?
It is known that the line graph of a Hamilton-connected graph G is also Hamilton-connected. Is μ(L(G)) Hamiltonconnected if L(G) is Hamilton-connected? [22].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.