Degree sum and connectivity conditions for dominating cycles

For a graph G, let σ_k(G) be the minimum degree sum of an independent set of k vertices. Ore showed that if G is a graph of order n?3 with σ₂(G)?n then G is hamiltonian. Let κ(G) be the connectivity of a graph G. Bauer, Broersma, Li and Veldman proved that if G is a 2-connected graph on n vertices with σ₃(G)?n+κ(G), then G is hamiltonian. On the other hand, Bondy showed that if G is a 2-connected graph on n vertices with σ₃(G)?n+2, then each longest cycle of G is a dominating cycle. In this paper, we prove that if G is a 3-connected graph on n vertices with σ₄(G)?n+κ(G)+3, then G contains a longest cycle which is a dominating cycle.     

On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v₁,…,v_kof G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K₄ and K_3,3. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K₄ and K_3,3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.     

Graph powers and k-ordered Hamiltonicity

It is known that if G is a connected simple graph, then G³ is Hamiltonian (in fact, Hamilton-connected). A simple graph is k-ordered Hamiltonian if for any sequence v₁, v₂,…,v_k of k vertices there is a Hamiltonian cycle containing these vertices in the given order. In this paper, we prove that if k?4, then G^⌊3k/2⌋-2 is k-ordered Hamiltonian for every connected graph G on at least k vertices. By considering the case of the path graph P_n, we show that this result is sharp. We also give a lower bound on the power of the cycle C_n that guarantees k-ordered Hamiltonicity.     

Heavy cycles in k-connected weighted graphs with large weighted degree sums

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. The weight of a cycle is defined as the sum of the weights of its edges. The weighted degree of a vertex is the sum of the weights of the edges incident with it. In this paper, we prove that: Let G be a k-connected weighted graph with k?2. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1), if G satisfies the following conditions: (1) The weighted degree sum of any k+1 pairwise nonadjacent vertices is at least m; (2) In each induced claw and each induced modified claw of G, all edges have the same weight. This generalizes an early result of Enomoto et al. on the existence of heavy cycles in k-connected weighted graphs.     

The Harary index of ordinary and generalized quasi-tree graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex v∈V(G) such that G?v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of all quasi-tree graphs of order n and characterized the corresponding extremal graphs. Moreover we defined the k-generalized quasi-tree graph to be a connected graph G with a subset V _k ?V(G) where |V _k |=k such that G?V _k is a tree. And we also determined the k-generalized quasi-tree graph of order n with maximal Harary index for all values of k and the extremal one with minimal Harary index for k=2.     

Iteratedk-line graphs

For integersk≥2, thek-line graph L_k(G) of a graph G is defined as a graph whose vertices correspond to the complete subgraphs onk vertices in G with two distinct vertices adjacent if the corresponding complete subgraphs have 1 common vertices inG. We define iteratedk-line graphs byL _k ⁿ (G) ?L _k (L _k ^n?1 (G), whereL _k ⁰ (G) ?G. In this paper the iterated behavior of thek-line graph operator is investigated. It turns out that the behavior is quite different fork = 2 (the well-known line graph case),k = 3, and k≥4.     

Path graphs

The concept of a line graph is generalized to that of a path graph. The path graph P_k(G) of a graph G is obtained by representing the paths P_k in G by vertices and joining two vertices whenever the corresponding paths P_k in G form a path P_k+1 or a cycle C_k. P₃-graphs are characterized and investigated on isomorphism and traversability. Trees and unicyclic graphs with hamiltonian P₃-graphs are characterized.     

The k-Dominating Graph

Given a graph G, the k-dominating graph of G, D _k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D _k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D _k (G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of D _k (G). In this paper we give conditions that ensure D _k (G) is connected.     

Edge-connectivity and edge-superconnectivity in sequence graphs

Given an integer k?1 and any graph G, the sequence graph S_k(G) is the graph whose set of vertices is the set of all walks of length k in G. Moreover, two vertices of S_k(G) are joined by an edge if and only if their corresponding walks are adjacent in G.In this paper we prove sufficient conditions for a sequence graph S_k(G) to be maximally edge-connected and edge-superconnected depending on the parity of k and on the vertex-connectivity of the original graph G.     

Neighbor Sum Distinguishing Index

We consider proper edge colorings of a graph G using colors of the set {1, . . . , k}. Such a coloring is called neighbor sum distinguishing if for any pair of adjacent vertices x and y the sum of colors taken on the edges incident to x is different from the sum of colors taken on the edges incident to y. The smallest value of k in such a coloring of G is denoted by ndi_Σ(G). In the paper we conjecture that for any connected graph G ≠ C ₅ of order n ≥ 3 we have ndi_Σ(G) ≤ Δ(G) + 2. We prove this conjecture for several classes of graphs. We also show that ndi_Σ(G) ≤ 7Δ(G)/2 for any graph G with Δ(G) ≥ 2 and ndi_Σ(G) ≤ 8 if G is cubic.     

Sums of powers of the degrees of a graph

For a graph G and k a real number, we consider the sum of the kth powers of the degrees of the vertices of G. We present some general bounds on this sum for various values of k.     

On k‐ordered graphs

Ng and Schultz [J Graph Theory 1 ( 6 ), 45–57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k‐ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k‐ordered Hamiltonian. We give sum of degree conditions for nonadjacent vertices and neighborhood union conditions that imply a graph is k‐ordered Hamiltonian. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 69–82, 2000     

Pancyclic and panconnected line graphs

Let G be a graph of order n ? 3. We prove that if G is k-connected (k ? 2) and the degree sum of k + 1 mutually independent vertices of G is greater than 1/3(k + 1)(n + 1), then the line graph L(G) of G is pancyclic. We also prove that if G is such that the degree sum of every 2 adjacent vertices is at least n, then L(G) is panconnected with some exceptions.     

Covering the vertices of a graph with cycles of bounded length

Let c_k(G) be the minimum number of elementary cycles of length at most k necessary to cover the vertices of a given graph G. In this work, we bound c_k(G) by a function of the order of G and its independence number.     

Multithreshold graphs

Multithreshold graphs are defined in terms of a finite sequence of real thresholds that break the real line into a set of regions, alternating between NO and YES. If real ranks can be assigned to the vertices of a graph in such a way that two vertices are adjacent iff the sum of their ranks lies in a YES region, then that graph is a multithreshold graph with respect to the given set of thresholds. If a graph can be represented with k or fewer thresholds, then it is k-threshold. The case of one threshold is the classical case introduced by Chvátal and Hammer. In this paper, we show for every graph G, there is a k such that G is k-threshold, and we exhibit graphs for which the required number of thresholds is linear in the order of the graph.     

On coloring graphs with locally small chromatic number

In 1973, P. Erdös conjectured that for eachkε2, there exists a constantc _k so that ifG is a graph onn vertices andG has no odd cycle with length less thanc _k n ^1/k , then the chromatic number ofG is at mostk+1. Constructions due to Lovász and Schriver show thatc _k , if it exists, must be at least 1. In this paper we settle Erdös’ conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.     

Geodesic-pancyclic graphs

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v in a graph G, denoted by d_G(u,v), is the number of edges in a u-v geodesic. A graph G with n vertices is panconnected if, for each pair of vertices u,v∈V(G) and for each integer k with d_G(u,v)?k?n-1, there is a path of length k in G that connects u and v. A graph G with n vertices is geodesic-pancyclic if, for each pair of vertices u,v∈V(G), every u-v geodesic lies on every cycle of length k satisfying max{2d_G(u,v),3}?k?n. In this paper, we study sufficient conditions of geodesic-pancyclic graphs. In particular, we show that most of the known sufficient conditions of panconnected graphs can be applied to geodesic-pancyclic graphs.     

Perfect k-line graphs and k-total graphs

The concept of the line graph can be generalized as follows. The k-line graph L_k(G) of a graph G is defined as a graph whose vertices are the complete subgraphs on k vertices in G. Two distinct such complete subgraphs are adjacent in L_k(G) if and only if they have in G k ? 1 vertices in common. The concept of the total graph can be generalized similarly. Then the Perfect Graph Conjecture will be proved for 3-line graphs and 3-total graphs. Moreover, perfect 3-line graphs are not contained in any of the known classes of perfect graphs. © 1993 John Wiley & Sons, Inc.     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

Dense Arbitrarily Vertex Decomposable Graphs

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n ₁, . . . , n _k ) of positive integers such that n ₁ + · · · + n _k = n there exists a partition (V ₁, . . . , V _k ) of the vertex set of G such that for each ${i \in \{1,\ldots,k\}}$ , V _i induces a connected subgraph of G on n _i vertices. The main result of the paper reads as follows. Suppose that G is a connected graph on n ≥ 20 vertices that admits a perfect matching or a matching omitting exactly one vertex. If the degree sum of any pair of nonadjacent vertices is at least n ? 5, then G is arbitrarily vertex decomposable. We also describe 2-connected arbitrarily vertex decomposable graphs that satisfy a similar degree sum condition.