Hamiltonian cycles through a linear forest in bipartite graphs

Pancyclic graphs and linear forests

We prove that a bipartite uniquely Hamiltonian graph has a vertex of degree 2 in each color class. As consequences, every bipartite Hamiltonian graph of minimum degree d has at least 21−dd! Hamiltonian cycles, and every bipartite Hamiltonian graph of minimum degree at least 4 and girth g has at least (3/2)g/8 Hamiltonian cycles. We indicate how the existence of more than one Hamiltonian cycle may lead to a general reduction method for Hamiltonian graphs.

Hamiltonian and long paths in bipartite graphs with connectivity

Hamiltonian and long cycles in bipartite graphs with connectivity

The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtained from C i − 1 by a switch for each i with 1 ≤ i ≤ t , where a switch is the replacement of a pair of edges u v and w z on a Hamiltonian cycle with the edges u w and v z of G, given that u w and v z did not appear on the cycle. We show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete, settling an open question posed by Ito et al. (2011) and van den Heuvel (2013). More precisely, we show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete for chordal bipartite graphs, strongly chordal split graphs, and bipartite graphs with maximum degree 6. Bipartite permutation graphs form a proper subclass of chordal bipartite graphs, and unit interval graphs form a proper subclass of strongly chordal graphs. On the positive side, we show that, for any two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time.

A Hamiltonian cycle in a graph G is a cycle which contains every vertex of G. The study of Hamiltonian cycle problem has a long history in graph theory and is a central theme. In general, it is NP-complete to decide whether a graph contains a Hamiltonian cycle. Thus researchers have been investigating sufficient conditions that guarantee the existence of a Hamiltonian cycle in a graph. There are many classic results along this line. For example, in 1952, Dirac showed that an n-vertex graph G with n ≥ 3 is Hamiltonian if δ(G) ≥ n. Chv´atal studied Hamiltonian cycles by considering graph toughness, a measure of resilience under the removal of vertices. Let t ≥ 0 be a real number and denote by c(G) the number of components of G. We say a graph G is t-tough if for each cut set S of G we have t · c(G − S) ≤ |S|. The toughness of a graph G, denoted τ (G), is the maximum value of t for which G is t-tough if G is non-complete, and is defined to be ∞ if G is complete. Chv´atal conjectured in 1973 the existence of some constant t such that all t-tough graphs with at least three vertices are Hamiltonian. While the conjecture has been proven for some special classes of graphs, it remains open in general. Supporting this conjecture of Chv´atal’s, in the first part of this thesis, we show that every 3-tough (P2 ∪ 3P1)-free graph with at least three vertices is Hamiltonian, where P2 ∪ 3P1 is the disjoint union of an edge and three isolated vertices. The notion of a 2-factor is a generalization of a Hamiltonian cycle, which consists of vertex disjoint cycles which together cover the vertices of G. Thus, a Hamiltonian cycle is just a 2-factor with exactly one cycle. It is known that every 2-tough graph with at least three vertices has a 2-factor. In graphs with restricted structures the toughness bound 2 can be improved. For example, it was shown that every 2K2-free 3/2-tough graph with at least three vertices has a 2-factor, and the toughness bound 3/2 is best possible. In viewing 2K2, the disjoint union of two edges, as a linear forest, in this thesis, for any linear forest R on 5, 6, or 7 vertices, we find the sharp toughness bound t such that every t-tough R-free graph

A cycle C in a graph G is a Hamilton cycle if C contains every vertex of G. Similarly, a path P in G is a Hamilton path if P contains every vertex of G. We say that G is Hamilton‐connected if for any pair of vertices, u and v of G, There exists a Hamilton path from u to v. If G is a bipartite graph with bipartition sets of equal size, and there is a Hamilton path from any vertex in one bipartition set to any vertex in the other, The n G is said to be Hamilton‐laceable. We present a proof showing that the n‐dimensional k‐ary butterfly graph, denoted BF(k, n), contains a Hamilton cycle. Then, we use this result in proving the stronger result that BF(k, n) is Hamilton‐laceable when n is even and Hamilton‐connected for odd values of n.

A collection L = P 1 ∪ P 2 ∪ · · · ∪ P t (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V (L)| = k is called a (k, t, s)-linear forest. A graph G is (k, t, s)ordered if for every (k, t, s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k, t, s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k, t, s)-ordered hamiltonian.

Hamiltonian cycles and dominating cycles passing through a linear forest

Tori are important fundamental interconnection networks for multiprocessor systems. Hamiltonian paths are important in information communication of multiprocessor systems, and Hamiltonian path embedding capability is an important aspect to determine if a network topology is suitable for a real application. In real systems, some links may have better performance. Therefore, when embedding Hamiltonian path into interconnection networks, it is desirable that these Hamiltonian paths would pass through the links with better performance. Given a two two-dimensional torus T (m, n) with m, n ≥ 5 odd, let L be a linear forest with at most two edges in T (m, n) and let u and v be two distinct vertices in T (m, n) such that none of the paths in L has u or v as internal node or both of them as end nodes. In this paper, we construct a hamiltonian path of T (m, n) between u and v passing through L.

Variations of coloring problems related to scheduling and discrete tomography

Scalable parallel algorithms for maximum matching and Hamiltonian circuit in convex bipartite graphs

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

Extremal problems on distance spectra of graphs

For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph. It is known from \cite{muller,garey} that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal bipartite graphs. A convex bipartite graph $G$ with bipartition $(X,Y)$ and an ordering $X=(x_1,\ldots,x_n)$, is a bipartite graph such that for each $y \in Y$, the neighborhood of $y$ in $X$ appears consecutively. $G$ is said to have convexity with respect to $X$. Further, convex bipartite graphs are a subclass of chordal bipartite graphs. In this paper, we present a necessary and sufficient condition for the existence of a Hamiltonian cycle in convex bipartite graphs and further we obtain a linear-time algorithm for this graph class. We also show that Chvatal's necessary condition is sufficient for convex bipartite graphs. The closely related problem is HAMILTONIAN PATH whose complexity is open in convex bipartite graphs. We classify the class of convex bipartite graphs as {\em monotone} and {\em non-monotone} graphs. For monotone convex bipartite graphs, we present a linear-time algorithm to output a Hamiltonian path. We believe that these results can be used to obtain algorithms for Hamiltonian path problem in non-monotone convex bipartite graphs. It is important to highlight (a) in \cite{keil,esha}, it is incorrectly claimed that Hamiltonian path problem in convex bipartite graphs is polynomial-time solvable by referring to \cite{muller} which actually discusses Hamiltonian cycle (b) the algorithm appeared in \cite{esha} for the longest path problem (Hamiltonian path problem) in biconvex and convex bipartite graphs have an error and it does not compute an optimum solution always. We present an infinite set of counterexamples in support of our claim.