Hamilton Paths in Dominating Graphs of Trees and Cycles

An edge-colored graph G is rainbow k-connected, if for every two vertices of G, there are k internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow k-connected coloring of G, $$rc_k(G)$$ , is the rainbow k-connection number of G. Let G and H be two connected graphs, where O is an orientation of G. Let $${\vec {e}}$$ be an oriented edge of H. The edge-comb product of G (under the orientation O) and H on $$\vec {e}$$ , $$G{}^o\rhd _{\vec {e}}H$$ , is a graph obtained by taking one copy of G and |E(G)| copies of H and identifying the i-th copy of H at the edge $${\vec {e}}$$ to the i-th edge of G, where the two edges have the same orientation. In this paper, we provide sharp lower and upper bounds for rainbow 2-connection numbers of edge-comb product of a cycle and a Hamiltonian graph. We also determine the rainbow 2-connection numbers of edge-comb product of a cycle with some graphs, i.e. complete graph, fan graph, cycle graph, and wheel graph.

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1 -factor contraction. This correspondence is most useful in the case when it induces a blue and red 2 -factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian I -graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian I -graphs follows from the fact that one can choose a 1 -factor in any I -graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4 , that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected I -graphs as a subfamily.

Finding a Hamiltonian cycle in a graph G = (V, E) is a well-known problem. The challenge of finding a Hamiltonian cycle that avoids these faults when faulty vertices or edges are present has been extensively studied. When the edge set of G is partitioned into k dimensions, the problem of dimension-balanced Hamiltonian cycles arises, where the Hamiltonian cycle uses approximately the same number of edges from each dimension (differing by at most one). This paper studies whether a dimension-balanced Hamiltonian cycle (DBH) exists in toroidal mesh graphs Tm,n when a single vertex or edge is faulty, called the one-fault DBH problem. We establish that Tm,n is one-fault DBH, except in the following cases: (1) both m and n are even; (2) one of m and n is 3, while the other satisfies mod 4 = 3 and is greater than 6; (3) one of m and n is odd, while the other satisfies mod 4 = 2. Additionally, this paper resolves a conjecture from prior literature, thereby providing a complete solution to the DBP problem on Tm,n.

We investigate hamiltonian properties of Pm × Cn, m ≥ 2 and even n ≥ 4, which is bipartite, in the presence of faulty vertices and/or edges. We show that Pm × Cn with n even is strongly hamiltonian-laceable if the number of faulty elements is one or less. When the number of faulty elements is two, it has a fault-free cycle of length at least mn - 2 unless both faulty elements are contained in the same partite vertex set; otherwise, it has a fault-free cycle of length mn - 4. A sufficient condition is derived for the graph with two faulty edges to have a hamiltonian cycle. By applying fault-hamiltonicity of Pm × Cn to a two-dimensional torus Cm × Cn, we obtain interesting hamiltonian properties of a faulty Cm × Cn.

We initialize a series of papers deepening the understanding of algorithmic properties of the width parameter maximum induced matching width (mim-width) of graphs. In this first volume we provide the first polynomial-time algorithms on graphs of bounded mim-width for problems that are not locally checkable. In particular, we givenO(w)-time algorithms on graphs of mim-width at most w, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and H-Induced Topological Minor for fixed H. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs, Convex graphs, k-Trapezoid, Circulark-Trapezoid, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k. We contrast these positive results to the fact that problems about finding long non-induced paths remain hard on graphs of bounded mim-width: We show that Hamiltonian Cycle (and hence Hamiltonian Path) is NP-hard on graphs of linear mim-width 1; this further hints at the expressive power of the mim-width parameter.

Cubic bipartite cyclic 4-connected graphs without Hamiltonian circuits

Hamiltonian cycle problems play a central role in graph theory and have wide-ranging applications in network-on-chip architectures, interconnection networks, and large-scale parallel systems. When a network contains faulty nodes or faulty links, the feasibility of certain paths becomes restricted, making the construction of Hamiltonian cycles substantially more difficult and increasingly important for ensuring reliable communication. A dimension-balanced Hamiltonian cycle is a special type of cycle that maintains an even distribution of edges across multiple dimensions of a network. Its directed counterpart extends this idea to symmetric directed networks by balancing the number of edges used in each positive and negative direction. Such cycles are desirable because they support uniform traffic distribution and reduce communication contention in practical systems. Previous research has examined the existence of directed dimension-balanced Hamiltonian cycles in directed toroidal mesh networks and has shown that some configurations permit directed dimension-balanced Hamiltonian cycles while others do not. Building on this foundation, this paper investigates the fault-tolerant properties of such networks by analyzing whether directed dimension-balanced Hamiltonian cycles still exist when a single vertex (node) or a single edge (link) is faulty. Our results extend the current understanding of Hamiltonian robustness in symmetric directed networks.

Algorithms to count paths and cycles

We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hubert space and creation operators that are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a 'higher rank semigroupoid algebra'. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular, the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs and characterize semisimplicity in terms of the underlying graph.

Circulant graphs are an extremely well-studied subclass of regular graphs, partially because they model many practical computer network topologies It has long been known that the number of spanning trees in n-node circulant graphs with constant jumps satisfies a recurrence relation in n For the non-constant-jump case, i.e., where some jump sizes can be functions of the graph size, only a few special cases such as the Mobius ladder had been studied but no general results were known. In this note we show how that the number of spanning trees for all classes of n node circulant graphs satisfies a recurrence relation in n even when the jumps are non-constant (but linear) in the graph size The technique developed is very general and can be used to show that many other structures of these circulant graphs, e.g., number of Hamiltonian Cycles, Eulerian Cycles, Eulerian Orientations, etc., also satisfy recurrence relations. The technique presented for deriving the recurrence relations is very mechanical and, for circulant graphs with small jump parameters, can easily be quickly implemented on a computer We illustrate this by deriving recurrence relations counting all of the structures listed above for various circulant graphs.

The coloring reconfiguration graph has as its vertex set all the proper ‐colorings of , and two vertices in are adjacent if their corresponding ‐colorings differ on a single vertex. Cereceda conjectured that if an ‐vertex graph is ‐degenerate and , then the diameter of is . Bousquet and Heinrich proved that if is planar and bipartite, then the diameter of is . (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when is planar and has no 3‐cycles. As a partial solution to this problem, we show that the diameter of is for every planar graph with no 3‐cycles and no 5‐cycles.

The dominating graph of a graph G has as its vertices all dominating sets of G, with an edge between two dominating sets if one can be obtained from the other by adding or deleting a single vertex of G. This is an example of a reconfiguration graph. This paper gives a brief introduction to the study of reconfiguration of dominating sets, and to the dominating graph. We highlight some previous results and present some new work. In particular we give new results on the existence of Hamilton paths in the dominating graph.

For a simple connected graph G, let c : V(G) → ℕ be a vertex coloring of G, where adjacent vertices may be colored the same. The neighborhood color set of a vertex v, denoted by NC(v), is the set of colors of the neighbors of v. The coloring c is called a set coloring provided that NC(u) ≠ NC(v) for every pair of adjacent vertices u and v of G. The minimum number of colors needed for a set coloring of G is referred to as the set chromatic number of G and is denoted by χs(G). In this work, the set chromatic number of graphs is studied in relation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles, paths, star graphs, double-star graphs, and some trees of height 2. Moreover, we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally, we develop an algorithm for constructing an optimal set coloring of the middle graph of trees of height 2 under some assumptions.

The Jordan centre of a graph is defined as a vertex whose maximum distance to other nodes in the graph is minimal, and it finds applications in facility location and source detection problems. We study properties of the Jordan centre in the case of random growing trees. In particular, we consider a regular tree graph on which an infection starts from a root node and then spreads along the edges of the graph according to various random spread models. For the Independent Cascade (IC) model and the discrete Susceptible Infected (SI) model, both of which are discrete-time models, we show that as the infected subgraph grows with time, the Jordan centre persists on a single vertex after a finite number of timesteps. As a corollary of our results, we also establish that the distance between the Jordan centre and the infection source (root node) is finite. Finally, we also study the continuous-time version of the SI model and bound the maximum distance between the Jordan centre and the root node at any time.

Recent results from Chih-Scull show that the ×-homotopy relation on finite graphs can be expressed via a sequence of “spider moves” which shift a single vertex at a time. In this paper we study a “spider web graph” which encodes exactly these spider moves between graph homomorphisms. We show how composition of graph homomorphisms relate to the spider web, study the components of spider webs for bipartite and tree graphs, andfinish by giving an explicit description of the spider web for homomorphisms from a bipartite graph to a star graph.