Distance-transitive representations of the symmetric groups

Let Γ be a graph and G ≤ Aut(Γ). The group G is said to act distance-transitively on Γ if, for any vertices x, y, u, v such that ∂(x, y) = ∂(u, v), there is an element g ϵ G mapping x into u and y into v. If G acts distance-transitively on Γ then the permutation group induced by the action of G on the vertex set of Γ is called the distance-transitive representation of G. In the paper all distance-transitive representations of the symmetric groups S_n are classified. Moreover, all pairs (G, Γ) such that G acts distance-transitively on Γ and G = S_n for some n are described. The classification problem for these pairs was posed by N. Biggs (Ann. N.Y. Acad. Sci. 319 (1979), 71–81). The problem is closely related to the general question about distance-transitive graphs with given automorphism group.     

Graphs with small boundary

For a pair of vertices x and y in a graph G, we denote by d_G(x,y) the distance between x and y in G. We call x a boundary vertex of y if x and y belong to the same component and d_G(y,v)?d_G(y,x) for each neighbor v of x in G. A boundary vertex of some vertex is simply called a boundary vertex, and the set of boundary vertices in G is called the boundary of G, and is denoted by B(G).In this paper, we investigate graphs with a small boundary. Since a pair of farthest vertices are boundary vertices, |B(G)|?2 for every connected graph G of order at least two. We characterize the graphs with boundary of order at most three. We cannot give a characterization of graphs with exactly four boundary vertices, but we prove that such graphs have minimum degree at most six. Finally, we give an upper bound to the minimum degree of a connected graph G in terms of |B(G)|.     

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.     

Diameter and connectivity 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 given graph G, X(G), is defined to have vertices the arcs of G. Two arcs uv,xy are adjacent in X(G) if and only if (v,u,x,y) is a 3-arc of G. This notion was introduced in recent studies of arc-transitive graphs. In this paper we study diameter and connectivity of 3-arc graphs. In particular, we obtain sharp bounds for the diameter and connectivity of X(G) in terms of the corresponding invariant of G.     

Isometric cycles,cutsets, and crowning of bridged graphs

A graph G is bridged if every cycle C of length at least 4 has vertices x,y such that d_G(x,y) < d_C(x,y). A cycle C is isometric if d_G(x,y) = d_C(x,y) for all x,y ∈ V(C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We introduce a “crowning” construction to enlarge bridged graphs. We use this to construct examples showing that for every connected simple graph H with girth at least 6 (including trees), there exists a bridged graph G such that G has a unique minimum cutset S and that G[S] = H. This provides counterexamples to Hahn's conjecture that d_G(u,v) ≤ 2 when u and v lie in a minimum cutset in a bridged graph G. We also study the convexity of cutsets in bridged graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 161–170, 2003     

Radio number for trees

Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f:V(G)→{0,1,2,…,k} with the following satisfied for all vertices u and v: |f(u)-f(v)|?diam(G)-d_G(u,v)+1, where d_G(u,v) is the distance between u and v. We prove a lower bound for the radio number of trees, and characterize the trees achieving this bound. Moreover, we prove another lower bound for the radio number of spiders (trees with at most one vertex of degree more than two) and characterize the spiders achieving this bound. Our results generalize the radio number for paths obtained by Liu and Zhu.     

A new sufficient condition for Hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u),d(v)}≥n/2 for each pair of vertices u and v with distance d(u,v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-connected graph and |N(u)∪N(v)|+δ(G)≥n for each pair of nonadjacent vertices u,v∈V(G), then G is Hamiltonian. This paper generalizes the above results when G is 3-connected. We show that if G is a 3-connected graph of order n and max{|N(x)∪N(y)|+d(u),|N(w)∪N(z)|+d(v)}≥n for every choice of vertices x,y,u,w,z,v such that d(x,y)=d(y,u)=d(w,z)=d(z,v)=d(u,v)=2 and where x,y and u are three distinct vertices and w,z and v are also three distinct vertices (and possibly |{x,y}∩{w,z}| is 1 or 2), then G is Hamiltonian.     

Induced path transit function, monotone and Peano axioms

The induced path transit function J(u,v) in a graph consists of the set of all vertices lying on any induced path between the vertices u and v. A transit function J satisfies monotone axiom if x,y∈J(u,v) implies J(x,y)⊆J(u,v). A transit function J is said to satisfy the Peano axiom if, for any u,v,w∈V,x∈J(v,w), y∈J(u,x), there is a z∈J(u,v) such that y∈J(w,z). These two axioms are equivalent for the induced path transit function of a graph. Planar graphs for which the induced path transit function satisfies the monotone axiom are characterized by forbidden induced subgraphs.     

A study 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 the arcs of G as vertices such that two arcs uv,xy are adjacent if and only if (v,u,x,y) is a 3-arc of G. In this paper, we study the independence, domination and chromatic numbers of 3-arc graphs and obtain sharp lower and upper bounds for them. We introduce a new notion of arc-coloring of a graph in studying vertex-colorings of 3-arc graphs.     

Integers with dense divisors

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e^−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x⁻¹D(x,y,1), for fixed u. This leads to a new estimate for d(u).     

On the Existence of a Long Path Between Specified Vertices in a 2-Connected Graph

 Let G be a 2-connected graph with maximum degree Δ (G)≥d, and let x and y be distinct vertices of G. Let W be a subset of V(G)−{x, y} with cardinality at most d−1. Suppose that max{d _G(u), d _G(v)}≥d for every pair of vertices u and v in V(G)−({x, y}∪W) with d _G(u,v)=2. Then x and y are connected by a path of length at least d−|W|. Received: February 5, 1998 Revised: April 13, 1998     

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 Chartrand-Schuster conjecture: Graphs with unique distance trees are regular

Let G be a finite connected graph with no cut vertex. A distance tree T is a spanning tree of G which further satisfies the condition that for some vertex v, d_G(v, u) = d_T(v, u) for all u, where d_G(v, u) denotes the distance of u from v in the graph G. The conjecture that if all distance trees of G are isomorphic to each other then G is a regular graph, is settled affirmatively. The conjecture was made by Chartrand and Schuster.     

On Properties of a Graph that Depend on its Distance Function

If G is a connected graph with distance function d, then by a step in G is meant an ordered triple (u, x, v) of vertices of G such that d(u, x) = 1 and d(u, v) = d(x, v) + 1. A characterization of the set of all steps in a connected graph was published by the present author in 1997. In Section 1 of this paper, a new and shorter proof of that characterization is presented. A stronger result for a certain type of connected graphs is proved in Section 2.     

Competitively Tight Graphs

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph G is related to the edge clique cover number θ _E (G) of the graph G via θ _E (G) ? |V(G)| + 2 ≤ k(G) ≤ θ _E (G). We first show that for any positive integer m satisfying 2 ≤ m ≤ |V(G)|, there exists a graph G with k(G) = θ _E (G) ? |V(G)| + m and characterize a graph G satisfying k(G) = θ _E (G). We then focus on what we call competitively tight graphs G which satisfy the lower bound, i.e., k(G) = θ _E (G) ? |V(G)| + 2. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.     

Almost 2-Homogeneous Graphs and Completely Regular Quadrangles

Many known distance-regular graphs have extra combinatorial regularities: One of them is t-homogeneity. A bipartite or almost bipartite distance-regular graph is 2-homogeneous if the number γ _i  = |{x | ∂(u, x) = ∂(v, x) = 1 and ∂(w, x) = i − 1}| (i = 2, 3,..., d) depends only on i whenever ∂(u, v) = 2 and ∂(u, w) = ∂(v, w) = i. K. Nomura gave a complete classification of bipartite and almost bipartite 2-homogeneous distance-regular graphs. In this paper, we generalize Nomura’s results by classifying 2-homogeneous triangle-free distance-regular graphs. As an application, we show that if Γ is a distance-regular graph of diameter at least four such that all quadrangles are completely regular then Γ is isomorphic to a binary Hamming graph, the folded graph of a binary Hamming graph or the coset graph of the extended binary Golay code of valency 24. We also consider the case Γ is a parallelogram-free distance-regular graph. This research was partially supported by the Grant-in-Aid for Scientific Research (No.17540039), Japan Society of the Promotion of Science.     

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.     

Characterization of projective graphs

We denote the distance between vertices x and y of a graph by d(x, y), and p_ij(x, y) = ∥ {z : d(x, z) = i, d(y, z) = j} ∥. The (s, q, d)-projective graph is the graph having the s-dimensional subspaces of a d-dimensional vector space over GF(q) as vertex set, and two vertices x, y adjacent iff $\text{dim}\text{(x ? y) = s ? 1}$. These graphs are regular graphs. Also, there exist integers λ and μ > 4 so that μ is a perfect square, p₁₁(x, y) = λ whenever d(x, y) = 1, and p₁₁(x, y) = μ whenever d(x, y) = 2. The (s, q, d)-projective graphs where $\text{2d}\text{3}\text{≤ s < d ? 2}$ and $\text{(s, q, d) ≠ (}\text{2d}\text{3}\text{, 2, d)}$, are characterized by the above conditions together with the property that there exists an integer r satisfying certain inequalities.     

Distance-regular graphs with girth 3 or 4: I

We study the structure of a distance-regular graph Γ with girth 3 or 4. First, we find some relationships among the intersection numbers of Γ when Γ contains a cycle {u₁, u₂, u₃, u₄} with ?(u₁, u₃) = ?(u₂, u₄) = 2. These relationships imply the diameter d, valency k, and intersection numbers a₁ and c_d of Γ are related by $\text{d ≤}\text{(k + c}{}_{\mathrm{d}}\text{)}\text{(a}{}_1\text{+ 2)}$. Next, we show subgraphs induced by vertex neighbourhoods in distance-regular graphs where cycles mentioned above do not exist are related to certain strongly regular graphs.     

The induced path function, monotonicity and betweenness

The geodesic interval function I of a connected graph allows an axiomatic characterization involving axioms on the function only, without any reference to distance, as was shown by Nebeský [20]. Surprisingly, Nebeský [23] showed that, if no further restrictions are imposed, the induced path function J of a connected graph G does not allow such an axiomatic characterization. Here J(u,v) consists of the set of vertices lying on the induced paths between u and v. This function is a special instance of a transit function. In this paper we address the question what kind of restrictions could be imposed to obtain axiomatic characterizations of J. The function J satisfies betweenness if w∈J(u,v), with w≠u, implies u∉J(w,v) and x∈J(u,v) implies J(u,x)⊆J(u,v). It is monotone if x,y∈J(u,v) implies J(x,y)⊆J(u,v). In the case where we restrict ourselves to functions J that satisfy betweenness, or monotonicity, we are able to provide such axiomatic characterizations of J by transit axioms only. The graphs involved can all be characterized by forbidden subgraphs.