Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups

In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph Γ is n‐HC‐extendable if it contains a path of length n and if every such path is contained in some Hamilton cycle of Γ. Similarly, Γ is weakly n‐HP‐extendable if it contains a path of length n and if every such path is contained in some Hamilton path of Γ. Moreover, Γ is strongly n‐HP‐extendable if it contains a path of length n and if for every such path P there is a Hamilton path of Γ starting with P. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2‐HC‐extendable and a complete classification of 3‐HC‐extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4‐HP‐extendable. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

A note on the girth‐doubling construction for polygonal graphs

A near‐polygonal graph is a graph Γ which has a set ?? of m‐cycles for some positive integer m such that each 2‐path of Γ is contained in exactly one cycle in ??. If m is the girth of Γ then the graph is called polygonal. Given a polygonal graph Γ of valency r and girth m, Archdeacon and Perkel proved the existence of a polygonal graph Γ₂ of valency r and girth 2m. We will show that this construction can be extended to one that yields a polygonal graph Γ₃ of valency r and girth 3m, but that making the cycles any longer with this construction does not yield a polygonal graph. We also show that if Aut(Γ) is 2‐arc transitive, so is Aut(Γ_k) for k = 2, 3. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 246‐254, 2011     

Even and odd holes in cap‐free graphs

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β‐perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even‐signable. Graphs that can be signed so that every triangle is odd and every triangle is odd and every hole is odd are called odd‐signable. We derive from a theorem due to Truemper co‐NP characterizations of even‐signable and odd‐signable graphs. A graph is strongly even‐signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even‐signable graph is even‐signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd‐signable. Every strongly odd‐signable graph is odd‐signable. We give co‐NP characterizations for both strongly even‐signable and strongly odd‐signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (cap‐free graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well‐studied subclass of cap‐free graphs. If a graph is strongly even‐signable or strongly odd‐signable, then it is cap‐free. In fact, strongly even‐signable graphs are those cap‐free graphs that are even‐signable. From our decomposition theorem, we derive decomposition results for strongly odd‐signable and strongly even‐signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 289–308, 1999     

Paths extendable to cycles

Let k be a positive integer, and S a nonempty set of positive integers. Suppose that G is a connected graph containing a path of length k, and that each path P of length k in G is contained in some cycle C(P) of length s ∈ S. We prove that every path of length less than k can be extended to a path of length k in G. This result answers conjectures of Entringer and Reid regarding when certain paths may be extended to cycles.     

Cycle-perfect graphs are perfect

The cycle graph of a graph G is the edge intersection graph of the set of all the induced cycles of G. G is called cycle-perfect if G and its cycle graph have no chordless cycles of odd length at least five. We prove the statement of the title. © 1996 John Wiley & Sons, Inc.     

Infinite paths in planar graphs IV,dividing cycles

Nash‐Williams conjectured that a 4‐connected infinite planar graph contains a spanning 2‐way infinite path if, and only if, the deletion of any finite set of vertices results in at most two infinite components. In this article, we prove this conjecture for graphs with no dividing cycles and for graphs with infinitely many vertex disjoint dividing cycles. A cycle in an infinite plane graph is called dividing if both regions of the plane bounded by this cycle contain infinitely many vertices of the graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 173–195, 2006     

Trivalent 2-Arc Transitive Graphs of Type {G^1_2} are Near Polygonal

A connected graph Σ of girth at least four is called a near n-gonal graph with respect to E, where n ≥  4 is an integer, if E is a set of n-cycles of Σ such that every path of length two is contained in a unique member of E. It is well known that connected trivalent symmetric graphs can be classified into seven types. In this note we prove that every connected trivalent G-symmetric graph S 1 K₄{\Sigma \neq K_4} of type G¹₂{G^1_2} is a near polygonal graph with respect to two G-orbits on cycles of Σ. Moreover, we give an algorithm for constructing the unique cycle in each of these G-orbits containing a given path of length two.     

Covering and Euler cycles on non-oriented graphs

A covering cycle is a closed path that traverses each edge of a graph at least once. Two cycles are equivalent if one is a cyclic permutation of the other. We compute the number of equivalence classes of non-periodic covering cycles of given length in a non-oriented connected graph. A special case is the number of Euler cycles (covering cycles that cover each edge of the graph exactly once) in the non-oriented graph. We obtain an identity relating the numbers of covering cycles of any length in a graph to a product of determinants.     

A contribution to queens graphs: A substitution method

A graph G is a queens graph if the vertices of G can be mapped to queens on the chessboard such that two vertices are adjacent if and only if the corresponding queens attack each other, i.e. they are in horizontal, vertical or diagonal position.We prove a conjecture of Beineke, Broere and Henning that the Cartesian product of an odd cycle and a path is a queens graph. We show that the same does not hold for two odd cycles. The representation of the Cartesian product of an odd cycle and an even cycle remains an open problem.We also prove constructively that any finite subgraph of the rectangular grid or the hexagonal grid is a queens graph.Using a small computer search we solve another conjecture of the authors mentioned above, saying that K_3,4 minus an edge is a minimal non-queens graph.     

Closure and stable Hamiltonian properties in claw‐free graphs

In the class of k‐connected claw‐free graphs, we study the stability of some Hamiltonian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e., for any of these properties there is an infinite family of graphs G_k of arbitrarily high connectivity k such that the closure of G_k has the property while the graph G_k does not); (ii) traceability is a stable property even for k = 1; (iii) homogeneous traceability is not stable for k = 2 (although it is stable for k = 7). The article is concluded with several open questions concerning stability of homogeneous traceability and Hamiltonian connectedness. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 30–41, 2000     

An efficient representation of Benes networks and its applications

The most popular bounded-degree derivative network of the hypercube is the butterfly network. The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly—like architectures. We identify a new topological representation of butterfly and Benes networks.The minimum metric dimension problem is to find a minimum set of vertices of a graph G(V,E) such that for every pair of vertices u and v of G, there exists a vertex w with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. It is NP-hard in the general sense. We show that it remains NP-hard for bipartite graphs. The algorithmic complexity status of this NP-hard problem is not known for butterfly and Benes networks, which are subclasses of bipartite graphs. By using the proposed new representations, we solve the minimum metric dimension problem for butterfly and Benes networks. The minimum metric dimension problem is important in areas such as robot navigation in space applications.     

Even Cycles in Graphs with Many Odd Cycles

It will be shown that if G is a graph of order n which contains a triangle, a cycle of length n or n−1 and at least cn odd cycles of different lengths for some positive constant c, then there exists some positive constant k=k(c) such that G contains at least kn ^1/6 even cycles of different lengths. Other results on the number of even cycle lengths which appear in graphs with many different odd length cycles will be given. Received: October 15, 1997     

Weakly pancyclic graphs

In generalizing the concept of a pancyclic graph, we say that a graph is “weakly pancyclic” if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it is pancyclic. This sheds some light on the content of a famous metaconjecture of Bondy. From the main result of this paper it follows that 2-connected nonbipartite graphs of sufficiently large order n with minimum degree exceeding 2n/7 are weakly pancyclic; and that graphs with minimum degree at least n/4 + 250 are pancyclic, if they contain both a triangle and a hamiltonian cycle. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 141–176, 1998     

On inverse problems for the cycle graph operator

The cycle graph of a graph is the intersection graph of the edge set of all the induced cycles ofH. The main result of this paper is: A (K ₄−e)-free graph is a cycle graph if and only if it is a block graph where each vertex lies in a finite number of blocks. Some additional results are also given.     

Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length ? for all 3≤?≤n. In 1972, Erd?s proved that if G is a Hamiltonian graph on n>4k⁴ vertices with independence number k, then G is pancyclic. He then suggested that n=Ω(k²) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n>ck^7/3 suffices.     

Vertex partitions of r -edge-colored graphs

Let G be an edge-colored graph. The monochromatic tree partition problem is to find the minimum number of vertex disjoint monochromatic trees to cover the all vertices of G. In the authors’ previous work, it has been proved that the problem is NP-complete and there does not exist any constant factor approximation algorithm for it unless P = NP. In this paper the authors show that for any fixed integer r ≥ 5, if the edges of a graph G are colored by r colors, called an r-edge-colored graph, the problem remains NP-complete. Similar result holds for the monochromatic path (cycle) partition problem. Therefore, to find some classes of interesting graphs for which the problem can be solved in polynomial time seems interesting. A linear time algorithm for the monochromatic path partition problem for edge-colored trees is given. Supported by the National Natural Science Foundation of China, PCSIRT and the “973” Program.     

A note on path and cycle decompositions of graphs

In the study of decompositions of graphs into paths and cycles, the following questions have arisen: Is it true that every graph G has a smallest path (resp. path-cycle) decomposition P such that every odd vertex of G is the endpoint of exactly one path of P? This note gives a negative answer to both questions.     

Diameter bounds for altered graphs

The main question addressed in this article is the following: If t edges are removed from a (t + 1) edge-connected graph G having diameter D, how large can the diameter of the resulting graph be? (The diameter of a graph is the maximum, over all pairs of vertices, of the length of the shortest path joining those vertices.) We provide bounds on this value that imply that the maximum possible diameter of the resulting graph, for large D and fixed t, is essentially (t + 1) · D. The bulk of the proof consists of showing that, if t edges are added to an n-vertex path P_n, then the diameter of the resulting graph is at least (n/(t + 1)) - 1. Using a similar proof, we also show that if t edges are added to an n-vertex cycle C_n, then the least possible diameter of the resulting graph is (for large n) essentially n/(t + 2) when t is even and n/(t + 1) when t is odd. Examples are given in all these cases to show that there exist graphs for which the bounds are achieved. We also give results for the corresponding vertex deletion problem for general graphs. Such results are of interest, for example, when studying the potential effects of node or link failures on the performance of a communication network, especially for networks in which the maximum time-delay or signal degradation is directly related to the diameter of the network.     

Quasi-transitive digraphs

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interseting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, we show that every strongly connected quasi-transitive digraphs D on at least four vertices has two vertices v₁ and v₂ such that D – v_i is strongly connected for i = 1, 2. A result of tournaments on the existence of a pair of arc-disjoint in- and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore we characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 ≦ k ≦ |V(D)| ? 1 through every vertex and yet they are not hamiltonian. Finally we characterize pancyclic and vertex pancyclic quasi-transitive digraphs. © 1995, John Wiley & Sons, Inc.     

On DRC‐covering of Kn by cycles

This paper considers the cycle covering of complete graphs motivated by the design of survivable WDM networks, where the requests are routed on INF‐networks which are protected independently from each other. The problem can be stated as follows: for a given graph G, find a cycle covering of the edge set of K_n, where V(K_n) = V(G), such that each cycle in the covering satisfies the disjoint routing constraint (DRC7rpar;, relatively to G, which can be stated as follows: to each edge of K_n we associate in G a path and all the paths associated to the edges of a cycle of the covering must be vertex disjoint. Here we consider the case where G = C_n, a ring of size n and we want to minimize the number of cycles in the covering. We give optimal solutions for the problem as well as for variations of the problem, namely, its directed version and the case when the cycle length is fixed to 4. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 100–112, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10040