A New Class of Brittle Graphs

A graph G is perfectly orderable in the sense of Chvátal if there exists a linear order on the set of vertices of G such that no induced path with vertices a, b, c, d and edges ab, bc, cd has a < b and d < c. A perfectly orderable graph G is brittle if every induced subgraph of G contains a vertex which is either endpoint or midpoint of no induced path with three edges in G. We present a new class of brittle graphs by forbidden configurations.     

k-ordered Hamiltonian graphs

A hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Theorems by Dirac and Ore, presenting sufficient conditions for a graph to be hamiltonian, are generalized to k-ordered hamiltonian graphs. The existence of k-ordered graphs with small maximum degree is investigated; in particular, a family of 4-regular 4-ordered graphs is described. A graph G of order n ≥ 3 is k-hamiltonian-connected, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices, G contains a v₁-v_k hamiltonian path that encounters v₁, v₂,…, v_k in this order. It is shown that for k ≥ 3, every (k + 1)-hamiltonian-connected graph is k-ordered and a result of Ore on hamiltonian-connected graphs is generalized to k-hamiltonian-connected graphs. © 1997 John Wiley & Sons, Inc.     

Edge disjoint cycles in graphs

Ore proved in 1960 that if G is a graph of order n and the sum of the degrees of any pair of nonadjacent vertices is at least n, then G has a hamiltonian cycle. In 1986, Li Hao and Zhu Yongjin showed that if n ? 20 and the minimum degree δ is at least 5, then the graph G above contains at least two edge disjoint hamiltonian cycles. The result of this paper is that if n ? 2δ², then for any 3 ? l₁ ? l₂ ? ? ? l_k ? n, 1 = k = [(δ - 1)/2], such graph has K edge disjoint cycles with lengths l₁, l₂…l_k, respectively. In particular, when l₁ = l₂ = ? = l_k = n and k = [(δ - 1)/2], the graph contains [(δ - 1)/2] edge disjoint hamiltonian cycles.     

Dirac-type conditions for hamiltonian paths and cycles in 3-uniform hypergraphs

A hamiltonian path (cycle) in an n-vertex 3-uniform hypergraph is a (cyclic) ordering of the vertices in which every three consecutive vertices form an edge. For large n, we prove an analog of the celebrated Dirac theorem for graphs: there exists n₀ such that every n-vertex 3-uniform hypergraph H, n?n₀, in which each pair of vertices belongs to at least n/2−1 (⌊n/2⌋) edges, contains a hamiltonian path (cycle, respectively). Both results are easily seen to be optimal.     

Acyclic graphoidal covers and path partitions in a graph

An acyclic graphoidal cover of a graph G is a collection ψ of paths in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of an acyclic graphoidal cover of G is called the acyclic graphoidal covering number of G and is denoted by η_a. A path partition of a graph G is a collection P of paths in G such that every edge of G is in exactly one path in P. The minimum cardinality of a path partition of G is called thepath partition number of G and is denoted by π. In this paper we determine η_a and π for several classes of graphs and obtain a characterization of all graphs with Δ 4 and η_a = Δ − 1. We also obtain a characterization of all graphs for which η_a = π.     

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.     

On a connection between the existence ofk-trees and the toughness of a graph

A graphG has toughnesst(G) ift(G) is the largest numbert such that for any subsetS of the vertices ofG, the number of vertices inS is at leastt times the number of components ofG on deletion of the vertices inS, provided that there is then more than one component. Ak-tree of a connected graph is a spanning tree with maximum degree at mostk. We show here that if , withk 3, thenG has ak-tree. The notion of ak-tree generalizes the casek = 2 of a hamiltonian path, so that this result, as we discuss, may be of some interest in connection with Chvátal's conjecture that, for somet, every graph with toughness at leastt is hamiltonian.     

A note on the weakly convex and convex domination numbers of a torus

The distanced_G(u,v) between two vertices u and v in a connected graph G is the length of the shortest (u,v) path in G. A (u,v) path of length d_G(u,v) is called a (u,v)-geodesic. A set X⊆V is called weakly convex in G if for every two vertices a,b∈X, exists an (a,b)-geodesic, all of whose vertices belong to X. A set X is convex in G if for all a,b∈X all vertices from every (a,b)-geodesic belong to X. The weakly convex domination number of a graph G is the minimum cardinality of a weakly convex dominating set of G, while the convex domination number of a graph G is the minimum cardinality of a convex dominating set of G. In this paper we consider weakly convex and convex domination numbers of tori.     

Degree Graphs of Simple Linear and Unitary Groups

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ? cd (G). We determine the graph Δ(G) for the finite simple groups of types A _?(q) and ² A _? (q ²), that is, for the simple linear and unitary groups.     

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.     

An improvement of fraisse's sufficient condition for hamiltonian graphs

Let G be a k-connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that Σ_xiEX d(X\{X_i}) > s(n?1) – k[s/2] – i(X)[(s?1)/2] holds for every independetn set X ={x₀, x₁ ?x_s} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.     

Fixed-edge theorem for graphs with loops

Let G be an undirected graph without multiple edges and with a loop at every vertex—the set of edges of G corresponds to a reflexive and symmetric binary relation on its set of vertices. Then every edge-preserving map of the set of vertices of G to itself fixes an edge [{f(a), f(b)} = {a, b} for some edge (a, b) of G] if and only if (i) G is connected, (ii) G contains no cycles, and (iii) G contains no infinte paths. The proof is conerned with those subgraphs H of a graph G for which there is an edge-preserving map f of the set of vertices of G onto the set of vertices of H and satisfying f(a) = a for each vertex a of H.     

Degree Graphs of Simple Groups of Exceptional Lie Type

Abstract

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type.     

On Non-Essential Edges in 3-Connected Graphs

 An edge e in a simple 3-connected graph is deletable (simple-contractible) if the deletion G\e (contraction G/e) is both simple and 3-connected. Suppose a, b, and c are three non-negative integers. If there exists a simple 3-connected graph with exactly a edges which are deletable but not simple-contractible, exactly b edges which are simple-contractible but not deletable, and exactly c edges which are both deletable and simple-contractible, then we call the triple (a, b, c) realizable, and such a graph is said to be an (a, b, c)-graph. Tutte's Wheels Theorem says the only (0, 0, 0)-graphs are the wheels. In this paper, we characterize the (a, b, c) realizable triples for which at least one of a + b≤2, c=0, and c≥16 holds. Received: February 12, 1997 Revised: February 13, 1998     

Hamiltonicity of 3-Arc Graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. We prove that any connected 3-arc graph is hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are hamiltonian. As a corollary we obtain that any vertex-transitive graph which is isomorphic to the 3-arc graph of a connected arc-transitive graph of degree at least three must be hamiltonian. This confirms the conjecture, for this family of vertex-transitive graphs, that all vertex-transitive graphs with finitely many exceptions are hamiltonian. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative 3-arc graphs.     

A Class of 2-Colorable Orthogonal Double Covers of Complete Graphs by Hamiltonian Paths

K _n by a graph G is a collection ? of n spanning subgraphs of K _n , all isomorphic to G, such that any two members of ? share exactly one edge and every edge of K _n is contained in exactly two members of ?. In the 1980's Hering posed the problem to decide the existence of an ODC for the case that G is an almost-hamiltonian cycle, i.e. a cycle of length n−1. It is known that the existence of an ODC of K _n by a hamiltonian path implies the existence of ODCs of K _4n and K _16n , respectively, by almost-hamiltonian cycles. Horton and Nonay introduced 2-colorable ODCs and showed: If for n≥3 and a prime power q≥5 there are an ODC of K _n by a hamiltonian path and a 2-colorable ODC of K _q by a hamiltonian path, then there is an ODC of K _qn by a hamiltonian path. We construct 2-colorable ODCs of K _n and K _2n , respectively, by hamiltonian paths for all odd square numbers n≥9. Received: January 27, 2000     

SOLVABLE GROUPS WITH CHARACTER DEGREE GRAPHS HAVING 5 VERTICES AND DIAMETER 3

ABSTRACT

Let G be a finite group and cd(G) the character degrees of G. The degree graph Δ(G) of G is the graph whose vertices are the primes dividing degrees in cd(G), and there is an edge between p and q if pq divides some degree in cd(G). In this paper, we show that if Δ(G) has 5 vertices, then the diameter of Δ(G) is at most 2. This shows that the example in^[9] of a solvable group G where Δ(G) has diameter 3 has the fewest number of vertices possible.     

Diperfect graphs

Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ₁, μ₂, ..., μ_k such tht |μ_i ∩S|=1 for alli. Later, Gallai proved that in a directed graph, the maximum number of vertices in a path is at least equal to the chromatic number; here again, we do not know if there exists an optimal coloring (S ₁,S ₂, ...,S _k) and a path μ such that |μ ∩S _i|=1 for alli. In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable setS there exists a partition of the vertex set into paths which meet the stable set in only one point. Also: for every optimal coloring there exists a path which meets each color class in only one point. This suggests several conjecties similar to the perfect graph conjecture. Dedicated to Tibor Gallai on his seventieth birthday     

Algebraic Shifting of Finite Graphs

In the present article, for bipartite graphs and chordal graphs, their exterior algebraic shifted graph and their symmetric algebraic shifted graph are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out that for a ≥ 3 and b ≥ 3, the exterior algebraic shifted graph of the complete bipartite graph K _a,b of size a, b is different from the symmetric algebraic shifted graph of K _a,b . Second, we will show that the exterior algebraic shifted graph of any chordal graph G coincides with the symmetric algebraic shifted graph of G. In addition, it will be shown that the exterior algebraic shifted graph of any chordal graph G is equal to some combinatorial shifted graph of G.     

Spanning Connectivity of the Power of a Graph and Hamilton-Connected Index of a Graph

Let G = (V, E) be a connected graph. The hamiltonian index h(G) (Hamilton-connected index hc(G)) of G is the least nonnegative integer k for which the iterated line graph L ^k (G) is hamiltonian (Hamilton-connected). In this paper we show the following. (a) If |V(G)| ≥ k + 1 ≥ 4, then in G ^k , for any pair of distinct vertices {u, v}, there exists k internally disjoint (u, v)-paths that contains all vertices of G; (b) for a tree T,  h(T) ≤ hc(T) ≤ h(T) + 1, and for a unicyclic graph G,  h(G) ≤ hc(G) ≤ max{h(G) + 1, k′ + 1}, where k′ is the length of a longest path with all vertices on the cycle such that the two ends of it are of degree at least 3 and all internal vertices are of degree 2; (c) we also characterize the trees and unicyclic graphs G for which hc(G) = h(G) + 1.