The connectivities of locally finite primitive graphs

Let be an infinite, locally finite graph whose automorphism group is primitive on its vertex set. It is shown that the connectivity of cannot equal 2, but all other values 0, 1, 3, 4, ... are possible.     

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.     

Orthogonal representations over finite fields and the chromatic number of graphs

We study the relationship between the minimum dimension of an orthogonal representation of a graph over a finite field and the chromatic number of its complement. It turns out that for some classes of matrices defined by a graph the 3-colorability problem is equivalent to deciding whether the class defined by the graph contains a matrix of rank 3 or not. This implies the NP-hardness of determining the minimum rank of a matrix in such a class. Finally we give for any class of matrices defined by a graph that is interesting in this respect a reduction of the 3-colorability problem to the problem of deciding whether or not this class contains a matrix of rank equal to three.The author is financially supported by the Cooperation Centre Tilburg and Eindhoven Universities.     

The edmonds—Gallai decomposition for matchings in locally finite graphs

We show that the Edmonds—Gallai decomposition theorem for matchings in finite graphs generalizes to all locally finite graphs. C.N.R.S. Dedicated to Tibor Gallai on his seventieth birthday     

On directed zero-divisor graphs of finite rings

For an artinian ring R, the directed zero-divisor graph Γ(R) is connected if and only if there is no proper one-sided identity element in R. Sinks and sources are characterized and clarified for a finite ring R. Especially, it is proved that for any ring R, if there exists a source y in Γ(R) with y²=0, then |R|=4 and R={0,x,y,z}, where x and z are left identity elements and yx=0=yz. Such a ring R is also the only ring such that Γ(R) has exactly one source. This shows that Γ(R) cannot be a network for any finite or infinite ring R.     

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     

Quantifier‐eliminable locally finite graphs

We identify the locally finite graphs that are quantifier‐eliminable and their first order theories in the signature of distance predicates. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On unimodular graphs

We study graphs whose adjacency matrices have determinant equal to 1 or −1, and characterize certain subclasses of these graphs. Graphs whose adjacency matrices are totally unimodular are also characterized. For bipartite graphs having a unique perfect matching, we provide a formula for the inverse of the corresponding adjacency matrix, and address the problem of when that inverse is diagonally similar to a nonnegative matrix. Special attention is paid to the case that such a graph is unicyclic.     

On thefg-coloring of graphs

This paper introduces a new type of edge-coloring of multigraphs, called anfg-coloring, in which each color appears at each vertexv no more thanf(v) times and at each set of multiple edges joining verticesv andw no more thang(vw) times. The minimum number of colors needed tofg-color a multigraphG is called thefg-chromatic index ofG. Various upper bounds are given on thefg-chromatic index. One of them is a generalization of Vizing's bound for the ordinary chromatic index. Our proof is constructive, and immediately yields a polynomial-time algorithm tofg-color a given multigraph using colors no more than twice thefg-chromatic index.     

Subgraphs of colour-critical graphs

Some problems and results on the distribution of subgraphs in colour-critical graphs are discussed. In section 3 arbitrarily largek-critical graphs withn vertices are constructed such that, in order to reduce the chromatic number tok−2, at leastc _k n ² edges must be removed. In section 4 it is proved that a 4-critical graph withn vertices contains at mostn triangles. Further it is proved that ak-critical graph which is not a complete graph contains a (k−1)-critical graph which is not a complete graph.     

Preperfect graphs

We say that a vertexx of a graph is predominant if there exists another vertexy ofG such that either every maximum clique ofG containingy containsx or every maximum stable set containingx containsy. A graph is then called preperfect if every induced subgraph has a predominant vertex. We show that preperfect graphs are perfect, and that several well-known classes of perfect graphs are preperfect. We also derive a new characterization of perfect graphs.     

Colouring series-parallel graphs

We establish a minimax formula for the chromatic index of series-parallel graphs; and also prove the correctness of a greedy algorithm for finding a vertex-colouring of a series-parallel graph.     

r-Strong edge colorings of graphs

Let G be a graph and for any natural number r, denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, . Here we generalize this result and show that . Moreover, we show that if for any two vertices u and v with maximum degree d(u,v)?3, then . Also for any tree T with Δ(T)?3 we prove that . Finally, it is shown that for any graph G with no isolated edges, .     

Colorings and orientations of graphs

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: IfG is a directed graph with maximum outdegreed, and if the number of Eulerian subgraphs ofG with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a setS(v) ofd+1 colors for each vertexv ofG there is a legal vertex-coloring ofG assigning to each vertexv a color fromS(v).Research supported in part by a United States-Israel BSF Grant and by a Bergmann Memorial Grant.     

Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number (G) and the chromatic number (G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each >0 and eachm2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+. This answers another question asked by Abbott and Zhou.     

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.     

Unit-distance graphs,graphs on the integer lattice and a Ramsey type result

Summary Let (R ², 1) denote the graph withR ² as the vertex set and two vertices adjacent if and only if their Euclidean distance is 1. The problem of determining the chromatic number(R ², 1) is still open; however,(R ², 1) is known to be between 4 and 7. By a theorem of de Bruijn and Erdös, it is enough to consider only finite subgraphs of (R ², 1). By a recent theorem of Chilakamarri, it is enough to consider certain graphs on the integer lattice. More precisely, forr > 0, let (Z ²,r, ) denote a graph with vertex setZ ² and two vertices adjacent if and only if their Euclidean distance is in the closed interval [r – ,r + ]. A simple graph is faithfully -recurring inZ ² if there exists a real numberd > 0 such that, for arbitrarily larger, G is isomorphic to a subgraph of (Z ²,r, ) in which every pair of vertices are at least distancedr apart. Chilakamarri has shown that, ifG is a finite simple graph, thenG is isomorphic to a subgraph of (R ², 1) if and only ifG is faithfully -recurring inZ ². In this paper we prove that(Z ²,r, ) 5 for integersr 1. We also prove a Ramsey type result which states that for any integerr > 1, and any coloring ofZ ² either there exists a monochromatic pair of vertices with their distance in the closed interval [r – ,r + ] or there exists a set of three vertices closest to each other with three distinct colors.     

Independent sets ink-chromatic graphs

A graph is said to have propertyP _k if in eachk-colouring ofG using allk colours there arek independent vertices having all colours. An (unpublished) suggestion of P. Erdős is answered in the affirmative: For eachk≧3 there is a k-critical graph withP _k . With the aid of a construction of T. Gallaik-chromatic graphs (k≧7) withP _k orP _k+1 of arbitrarily high connectivity are obtained. The main result is: Eachk-chromatic graph (k≧3) of girth ≧6 hasP _k or is a circuit of length 7. Dedicated to Paul Erdős on his seventieth birthday     

Radius-edge-invariant and diameter-edge-invariant graphs

The eccentricity e(v) of v is the distance to a farthest vertex from v. The radius r(G) is the minimum eccentricity among the vertices of G and the diameter d(G) is the maximum eccentricity. For graph G−e obtained by deleting edge e in G, we have r(G−e)?r(G) and d(G−e)?d(G). If for all e in G, r(G−e)=r(G), then G is radius-edge-invariant. Similarly, if for all e in G, d(G−e)=d(G), then G is diameter-edge-invariant. In this paper, we study radius-edge-invariant and diameter-edge-invariant graphs and obtain characterizations of radius-edge-invariant graphs and diameter-edge-invariant graphs of diameter two.     

Minimal orientations of colour critical graphs

In 1966 T. Gallai asked whether every criticalk-chromatic graph possesses an orientation having just one directed path of lengthk–1. In this note we show that in general the answer is negative, but also that the answer is affirmative whenk5 and the graph has maximal degree at mostk.