A characterization of panconnected graphs satisfying a local ore-type condition

It is well known that a graph G of order p ≥ 3 is Hamilton-connected if d(u) + d(v) ≥ p + 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| + 1 for any path uwv with uv ∉ E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) ≤ k ≤ |V(G)| − 1, there is an x − y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k η {3,4}) there is a cycle of length k containing xy. Consequently G is panconnected (and also edge pancyclic) if and only if each edge of G belongs to a triangle and a quadrangle. Our results imply some results of Williamson, Faudree, and Schelp. © 1996 John Wiley & Sons, Inc.     

The strong chromatic index of a class of graphs

The strong chromatic index of a graph G is the minimum integer k such that the edge set of G can be partitioned into k induced matchings. Faudree et al. [R.J. Faudree, R.H. Schelp, A. Gyárfás, Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205-211] proposed an open problem: If G is bipartite and if for each edge xy∈E(G), d(x)+d(y)≤5, then sχ^′(G)≤6. Let H₀ be the graph obtained from a 5-cycle by adding a new vertex and joining it to two nonadjacent vertices of the 5-cycle. In this paper, we show that if G (not necessarily bipartite) is not isomorphic to H₀ and d(x)+d(y)≤5 for any edge xy of G then sχ^′(G)≤6. The proof of the result implies a linear time algorithm to produce a strong edge coloring using at most 6 colors for such graphs.     

Closures,cycles, and paths

In 1960 Ore proved the following theorem: Let G be a graph of order n. If d(u) + d(v)≥n for every pair of nonadjacent vertices u and v, then G is hamiltonian. Since then for several other graph properties similar sufficient degree conditions have been obtained, so‐called “Ore‐type degree conditions”. In [R. J. Faudree, R. H. Schelp, A. Saito, and I. Schiermeyer, Discrete Math 307 (2007), 873–877], Faudree et al. strengthened Ore's theorem as follows: They determined the maximum number of pairs of nonadjacent vertices that can have degree sum less than n (i.e. violate Ore's condition) but still imply that the graph is hamiltonian. In this article we prove that for some other graph properties the corresponding Ore‐type degree conditions can be strengthened as well. These graph properties include traceable graphs, hamiltonian‐connected graphs, k‐leaf‐connected graphs, pancyclic graphs, and graphs having a 2‐factor with two components. Graph closures are computed to show these results. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 314–323, 2012     

New Ore��s Type Results on Hamiltonicity and Existence of Paths of Given Length in Graphs

The well-known Ore??s theorem (see Ore in Am Math Mon 65:55, 1960), states that a graph G of order n such that d(x)?+?d(y)??? n for every pair {x, y} of non-adjacent vertices of G is Hamiltonian. In this paper, we considerably improve this theorem by proving that in a graph G of order n and of minimum degree ????? 2, if there exist at least n ? ?? vertices x of G so that the number of the vertices y of G non-adjacent to x and satisfying d(x)?+?d(y)??? n ? 1 is at most ?? ? 1, then G is Hamiltonian. We will see that there are graphs which violate the condition of the so called ??Extended Ore??s theorem?? (see Faudree et?al. in Discrete Math 307:873?C877, 2007) as well as the condition of Chvatál??s theorem (see Chvátal in J Combin Theory Ser B 12:163?C168, 1972) and the condition of the so called ??Extended Fan?? theorem?? (see Faudree et?al. in Discrete Math 307:873?C877, 2007), but satisfy the condition of our result, which then, only allows to conclude that such graphs are Hamiltonian. This will show the pertinence of our result. We give also a new result of the same type, ensuring the existence of a path of given length.     

All cycle‐complete graph Ramsey numbers r(Cm,K6)

The cycle‐complete graph Ramsey number r(C_m, K_n) is the smallest integer N such that every graph G of order N contains a cycle C_m on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erd?s, Faudree, Rousseau and Schelp that r(C_m, K_n) = (m ? 1) (n ? 1) + 1 for all m ≥ n ≥ 3 (except r(C₃, K₃) = 6). This conjecture holds for 3 ≤ n ≤ 5. In this paper we will present a proof for n = 6 and for all n ≥ 7 with m ≥ n² ? 2n. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 251–260, 2003     

A sparse graph almost as good as the complete graph on points inK dimensions

A setV ofn points ink-dimensional space induces a complete weighted undirected graph as follows. The points are the vertices of this graph and the weight of an edge between any two points is the distance between the points under someL _p metric. Let ε≤1 be an error parameter and letk be fixed. We show how to extract inO(n logn+ε ^−k log(1/ε)n) time a sparse subgraphG=(V, E) of the complete graph onV such that: (a) for any two pointsx, y inV, the length of the shortest path inG betweenx andy is at most (1+∈) times the distance betweenx andy, and (b)|E|=O(ε^−k n).     

On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Abstract

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic di?erence of the function f by the formula

Q_k f (x, y) := f (x + ky) + f (x ? ky) ? f (x + y) ? f (x ? y) ? 2(k² ? 1)f (y)

for all x, y ∈ G and for any integer k with k ≠ 1, ?1. In this paper, we achieve the general solution of equation Q_k f (x, y) = 0, after it, we show that if Q_k f is Lipschitz, then there exists a quadratic function K : G → E such that f ? K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].     

Non-Separating Paths in 4-Connected Graphs

In 1975, Lovász conjectured that for any positive integer k, there exists a minimum positive integer f(k) such that, for any two vertices x, y in any f(k)-connected graph G, there is a path P from x to y in G such that G–V(P) is k-connected. A result of Tutte implies f(1) = 3. Recently, f(2) = 5 was shown by Chen et al. and, independently, by Kriesell. In this paper, we show that f(2) = 4 except for double wheels.Received October 17, 2003     

Cycle and cocycle coverings of graphs

In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family ?? of at most n?1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and ε edges having cogirth g^*?3 and k(G) components, there is a family of at most ε?n+k(G) cocycles which cover the edges of G at least twice. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 270–284, 2010     

Abelian Group Gradings on Full Matrix Rings

For positive integers ? and n, several authors studied ?_?-gradings of the full matrix ring M _n (k) over a field k. In this article, we show that every (G × H)-grading of M _n (k) can be constructed by a pair of compatible G-grading and H-grading of M _n (k), where G and H are any finite groups. When G and H are finite cyclic groups, we characterize all (G × H)-gradings which are isomorphic to a good grading. Moreover, the results can be generalized for any finite abelian group grading of M _n (k).     

Distributing pairs of vertices on Hamiltonian cycles

Let G be a graph of order n with minimum degree δ(G)≥n/2+1. Faudree and Li(2012) conjectured that for any pair of vertices x and y in G and any integer 2≤k≤n/2, there exists a Hamiltonian cycle C such that the distance between x and y on C is k. In this paper, we prove that this conjecture is true for graphs of sufficiently large order. The main tools of our proof are the regularity lemma of Szemer′edi and the blow-up lemma of Koml′os et al.(1997).     

A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

Fan-type theorem for path-connectivity

A graphG is said to bek-path-connected if every pair of distinct vertices inG are joined by a path of length at leastk. We prove that if max{deg _G ^x , deg _G ^y }k for every pair of verticesx,y withd _G (x,y)=2 in a 2-connected graphG, whered _G (x,y) is the distance betweenx andy inG, thenG isk-path-connected.     

On weights of induced paths and cycles in claw‐free and K1,r‐free graphs

Let G be a K_1,r ‐free graph (r ≥ 3) on n vertices. We prove that, for any induced path or induced cycle on k vertices in G (k ≥ 2r − 1 or k ≥ 2r, respectively), the degree sum of its vertices is at most (2r − 2)(n − α) where α is the independence number of G. As a corollary we obtain an upper bound on the length of a longest induced path and a longest induced cycle in a K_1,r ‐free graph. Stronger bounds are given in the special case of claw‐free graphs (i.e., r = 3). Sharpness examples are also presented. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 131–143, 2001     

Linear arboricity for graphs with multiple edges

Akiyama, Exoo, and Harary conjectured that for any simple graph G with maximum degree Δ(G), the linear arboricity la(G) satisfies ?Δ(G)/2? ≦ la(G) ≦ ?(Δ(G) + 1)/2?. Here it is proved that if G is a loopless graph with maximum degree Δ(G) ≦ k and maximum edge multiplicity μ(G) ≦ k ? 2ⁿ⁺¹ + 1, where k ≧ 2^n?2, then la(G) ≦ k ? 2ⁿ. It is also conjectured that for any loopless graph G, ?Δ(G)/2? ≦ la(G) ≦ ?(Δ(G) + μ(G))/2?.     

Disjoint hamiltonian cycles in bipartite graphs

Let G=(X,Y) be a bipartite graph and define . Moon and Moser [J. Moon, L. Moser, On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163-165. MR 28 # 4540] showed that if G is a bipartite graph on 2n vertices such that , then G is hamiltonian, sharpening a classical result of Ore [O. Ore, A note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that , then G contains k edge-disjoint hamiltonian cycles. This extends the result of Moon and Moser and a result of R. Faudree et al. [R. Faudree, C. Rousseau, R. Schelp, Edge-disjoint Hamiltonian cycles, Graph Theory Appl. Algorithms Comput. Sci. (1984) 231-249].     

Preemptive Scheduling of Equal-Length Jobs in Polynomial Time

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

Path extendability of claw-free graphs

Let G be a connected, locally connected, claw-free graph of order n and x,y be two vertices of G. In this paper, we prove that if for any 2-cut S of G, S∩{x,y}=∅, then each (x,y)-path of length less than n-1 in G is extendable, that is, for any path P joining x and y of length h(<n-1), there exists a path P^′ in G joining x and y such that V(P)⊂V(P^′) and |P^′|=h+1. This generalizes several related results known before.     

Subpancyclicity of line graphs and degree sums along paths

A graph is called subpancyclic if it contains a cycle of length ? for each ? between 3 and the circumference of the graph. We show that if G is a connected graph on n?146 vertices such that d(u)+d(v)+d(x)+d(y)>(n+10/2) for all four vertices u,v,x,y of any path P=uvxy in G, then the line graph L(G) is subpancyclic, unless G is isomorphic to an exceptional graph. Moreover, we show that this result is best possible, even under the assumption that L(G) is hamiltonian. This improves earlier sufficient conditions by a multiplicative factor rather than an additive constant.