A Ramsey-type theorem for traceable graphs

A graph G is traceable if there is a path passing through all the vertices of G. It is proved that every infinite traceable graph either contains arbitrarily large finite chordless paths, or contains a subgraph isomorphic to graph A, illustrated in the text. A corollary is that every finitely generated infinite lattice of length 3 contains arbitrarily large finite fences. It is also proved that every infinite traceable graph containing no chordless four-point path contains a subgraph isomorphic to K_ω,ω. The versions of these results for finite graphs are discussed.     

Über eine eigenschaft lokalfiniter,unendlicher bäume

Let A be an arbitrary locally finite, infinite tree and assume that a graph G contains for every positive integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Then G contains infinitely many disjoint subgraphs each isomorphic to a subdivision of A. This sharpens a theorem of Halin [5], who proved the corresponding result for the case that A is a tree in which each vertex has degree not greater than 3.     

On a problem of R. Halin concerning infinite graphs II

For every countable, connected graph A containing no one-way infinite path the following is shown: Let G be an arbitrary graph which contains for every positive integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Then G also contains infinitely many disjoint subgraphs each isomorphic to a subdivision of A. In addition, corrections of errors are given that occur unfortunately in the forerunner of the present paper.     

On disjoint configurations in infinite graphs

For a graph A and a positive integer n, let nA denote the union of n disjoint copies of A; similarly, the union of ?₀ disjoint copies of A is referred to as ?₀A. It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all n??, but ?₀A is not a minor of G. This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G₁, G₂,… such that G_i is not a minor of G_j for all i ≠ j. In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G, if nA is a minor of G for all n ? ?, then ?₀A is a minor of G, too. If true, this would be a far‐reaching generalization of a classical result of R. Halin on families of disjoint one‐way infinite paths in graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222–229, 2002; DOI 10.1002/jgt.10016     

On the Metric Dimension of Infinite Graphs

A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs.     

Colorful polytopes and graphs

The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P _G of rank n, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P _G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayley graph, in particular the graphicahedra, as well as the flagadjacency polytopes and related monodromy polytopes associated with a given abstract polytope. The duals of certain families of colorful polytopes have been important in the topological study of colored triangulations and crystallization of manifolds.     

On end degrees and infinite cycles in locally finite graphs

We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kühn [4, 5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.     

Representing triangulated graphs in stars

A graphG is calledrepresentable in a tree T, ifG is isomorphic to the intersection graph of a family of subtrees ofT. In this paper those graphs are characterized which are representable in some subdivision of theK _1,n. In the finite case polynomial-time recognition algorithms of these graphs are given. But this concept can be generalized to essentially infinite graphs by using no more trees but ‘tree-like’ posets and representability of graphs in these posets.     

Reconstructing the degree sequence and the number of components of an infinite graph

We prove that the degree sequence of an infinite graph is reconstructible from its family of vertex-deleted subgraphs. Furthermore, as another result concerning the reconstruction of infinite graphs, we prove that the number c(G) of components of an infinite graph G is reconstructible if there is at least one vertex x in G with the following property: If x is deleted from G, then the component of G containing x splits into a finite number of components. In particular, this implies that c(G) is reconstructible if there is at least one vertex of finite degree in G.     

On the nullity of tricyclic graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G)?n-2 if G is a simple graph on n vertices and G is not isomorphic to nK₁. The extremal graphs attaining the upper bound n-2 and the second upper bound n-3 have been obtained. In this paper, the graphs with nullity n-4 are characterized. Furthermore the tricyclic graphs with maximum nullity are discussed.     

On large light graphs in families of polyhedral graphs

A graph H is said to be light in a family H of graphs if each graph G∈H containing a subgraph isomorphic to H contains also an isomorphic copy of H such that each its vertex has the degree (in G) bounded above by a finite number φ(H,H) depending only on H and H. We prove that in the family of all 3-connected plane graphs of minimum degree 5 (or minimum face size 5, respectively), the paths with certain small graphs attached to one of its ends are light.     

Duality of infinite graphs

Some basic results on duality of infinite graphs are established and it is proven that a block has a dual graph if and only if it is planar and any two vertices are separated by a finite edge cut. Also, the graphs having predual graphs are characterized completely and it is shown that if $\text{G}{}^1$ is a dual and predual graph of G, then G and $\text{G}{}^1$ can be represented as geometric dual graphs. The uniqueness of dual graphs is investigated, in particular, Whitney's 2-isomorphism theorem is extended to infinite graphs. Finally, infinite minimal cuts in dual graphs are studied and the characterization (in terms of planarity and separation properties) of the graphs having dual graphs satisfying conditions on the infinite cuts, as well, is included.     

Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

Representing edge intersection graphs of paths on degree 4 trees

Let P be a collection of nontrivial simple paths on a host tree T. The edge intersection graph of P, denoted by EPT(P), has vertex set that corresponds to the members of P, and two vertices are joined by an edge if and only if the corresponding members of P share at least one common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree if G=EPT(P) for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree network or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph.An undirected graph G is chordal if every cycle in G of length greater than 3 possesses a chord. Chordal graphs correspond to vertex intersection graphs of subtrees on a tree. An undirected graph G is weakly chordal if every cycle of length greater than 4 in G and in its complement possesses a chord. It is known that the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. Moreover, this provides an algorithm to reduce a given EPT representation of a weakly chordal EPT graph to an EPT representation on a degree 4 tree. Finally, we raise a number of intriguing open questions regarding related families of graphs.     

The lollipop graph is determined by its Q-spectrum

A graph G is said to be determined by its Q-spectrum if with respect to the signless Laplacian matrix Q, any graph having the same spectrum as G is isomorphic to G. The lollipop graph, denoted by H_n,p, is obtained by appending a cycle C_p to a pendant vertex of a path P_n−p. In this paper, it is proved that all lollipop graphs are determined by their Q-spectra.     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

Triangle-free partial graphs and edge covering theorems

In section 1 some lower bounds are given for the maximal number of edges ofa (p ? 1)- colorable partial graph. Among others we show that a graph on n vertices with m edges has a (p?1)-colorable partial graph with at least mT_n.p/(ⁿ₂) edges, where T_n.p denotes the so called Turán number. These results are used to obtain upper bounds for special edge covering numbers of graphs. In Section 2 we prove the following theorem: If G is a simple graph and μ is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by μ triangles and edges. In Section 3 related questions are examined.     

G-decomposition of Kn, where G has four vertices or less

The complete graph K_n, is said to have a G-decomposition if it is the union of edge disjoint subgraphs each isomorphic to G. The set of values of n for which K_n has a G-decomposition is determined if G has four vertices or less.     

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.