The Set of Circular Flow Numbers of Regular Graphs

This article determines the set of the circular flow numbers of regular graphs. Let be the set of the circular flow numbers of graphs, and be the set of the circular flow numbers of d‐regular graphs. If d is even, then . For it is known 6 that . We show that . Hence, the interval is the only gap for circular flow numbers of ‐regular graphs between and 5. Furthermore, if Tutte's 5‐flow conjecture is false, then it follows, that gaps for circular flow numbers of graphs in the interval [5, 6] are due for all graphs not just for regular graphs.     

Extremal Graphs With a Given Number of Perfect Matchings

Let denote the maximum number of edges in a graph having n vertices and exactly p perfect matchings. For fixed p, Dudek and Schmitt showed that for some constant when n is at least some constant . For , they also determined and . For fixed p, we show that the extremal graphs for all n are determined by those with vertices. As a corollary, a computer search determines and for . We also present lower bounds on proving that for (as conjectured by Dudek and Schmitt), and we conjecture an upper bound on . Our structural results are based on Lovász's Cathedral Theorem.     

Graphs of Large Linear Size Are Antimagic

Given a graph and a colouring , the induced colour of a vertex v is the sum of the colours at the edges incident with v. If all the induced colours of vertices of G are distinct, the colouring is called antimagic. If G has a bijective antimagic colouring , the graph G is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than K₂ are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least for some constant c; we improve on this result, proving the conjecture for graphs with average degree at least some constant d₀.     

Generating Nonisomorphic Quadrangular Embeddings of a Complete Graph

We construct (resp. ) index one current graphs with current group such that the current graphs have different underlying graphs and generate nonisomorphic orientable (resp. nonorientable) quadrangular embeddings of the complete graph , (resp. ).     

Finite 2‐Geodesic Transitive Graphs of Prime Valency

We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc transitive or the valency p satisfies , and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph with automorphism group and diameter 3.     

Minimum Vertex Degree Threshold for ‐tiling*

We prove that the vertex degree threshold for tiling (the 3‐uniform hypergraph with four vertices and two triples) in a 3‐uniform hypergraph on vertices is , where if and otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling.     

The Degree‐Diameter Problem for Claw‐Free Graphs and Hypergraphs

We study the degree‐diameter problem for claw‐free graphs and 2‐regular hypergraphs. Let be the largest order of a claw‐free graph of maximum degree Δ and diameter D. We show that , where , for any D and any even . So for claw‐free graphs, the well‐known Moore bound can be strengthened considerably. We further show that for with (mod 4). We also give an upper bound on the order of ‐free graphs of given maximum degree and diameter for . We prove similar results for the hypergraph version of the degree‐diameter problem. The hypergraph Moore bound states that the order of a hypergraph of maximum degree Δ, rank k, and diameter D is at most . For 2‐regular hypergraph of rank and any diameter D, we improve this bound to , where . Our construction of claw‐free graphs of diameter 2 yields a similar result for hypergraphs of diameter 2, degree 2, and any even rank .     

Partial Online List Coloring of Graphs

For a graph G, let be the maximum number of vertices of G that can be colored whenever each vertex of G is given t permissible colors. Albertson, Grossman, and Haas conjectured that if G is s‐choosable and , then . In this article, we consider the online version of this conjecture. Let be the maximum number of vertices of G that can be colored online whenever each vertex of G is given t permissible colors online. An analog of the above conjecture is the following: if G is online s‐choosable and then . This article generalizes some results concerning partial list coloring to online partial list coloring. We prove that for any positive integers , . As a consequence, if s is a multiple of t, then . We also prove that if G is online s‐choosable and , then and for any , .     

Equimatchable Graphs on Surfaces

A graph G is equimatchable if each matching in G is a subset of a maximum‐size matching and it is factor critical if has a perfect matching for each vertex v of G. It is known that any 2‐connected equimatchable graph is either bipartite or factor critical. We prove that for 2‐connected factor‐critical equimatchable graph G the graph is either or for some n for any vertex v of G and any minimal matching M such that is a component of . We use this result to improve the upper bounds on the maximum number of vertices of 2‐connected equimatchable factor‐critical graphs embeddable in the orientable surface of genus g to if and to if . Moreover, for any nonnegative integer g we construct a 2‐connected equimatchable factor‐critical graph with genus g and more than vertices, which establishes that the maximum size of such graphs is . Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2‐connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face‐width k. Finally, we prove that any d‐degenerate 2‐connected equimatchable factor‐critical graph has at most vertices, where a graph is d‐degenerate if every its induced subgraph contains a vertex of degree at most d.     

Circular Chromatic Indices of Regular Graphs

The circular chromatic index of a graph G, written , is the minimum r permitting a function such that whenever e and are adjacent. It is known that for any , there is a 3‐regular simple graph G with . This article proves the following results: Assume is an odd integer. For any , there is an n‐regular simple graph G with . For any , there is an n‐regular multigraph G with .     

Destroying Noncomplete Regular Components in Graph Partitions

We prove that if G is a graph and such that then can be partitioned into sets such that and contains no noncomplete ‐regular components for each . In particular, the vertex set of any graph G can be partitioned into sets, each of which induces a disjoint union of triangles and paths.     

Locally Symmetric Graphs of Girth 4

We classify the family of connected, locally symmetric graphs of girth 4 (finite and infinite). They are all regular, with the exception of the complete bipartite graph . There are, up to isomorphism, exactly four such k‐regular graphs for every , one for , two for , and exactly three for every infinite cardinal k. In the last paragraph, we consider locally symmetric graphs of girth >4.     

Perfect Matchings Avoiding Several Independent Edges in a Star‐Free Graph

In Aldred and Plummer (Discrete Math 197/198 (1999) 29–40) proved that every m‐connected ‐free graph of even order has a perfect matching M with and , where F₁ and F₂ are prescribed disjoint sets of independent edges with and . It is known that if l satisfies , then the star‐free condition in the above result is best possible. In this paper, for , we prove a refinement of the result in which the condition is replaced by the weaker condition that G is ‐free (note that the new condition does not depend on l). We also show that if m is even and either or , then for m‐connected graphs G with sufficiently large order, one can replace the condition by the still weaker condition that G is ‐free. The star‐free conditions in our results are best possible.     

Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs II: Voltage Graph Constructions and Applications

In an earlier article the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two‐part series, we extend those results to orientable surfaces for all . In part II, a voltage graph construction is presented for building embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that p is prime, completing the proof started by part I (which covers the case ) that there exists an orientable hamilton cycle embedding of for all , . These embeddings are then used to determine the genus of several families of graphs, notably for and, in some cases, for .     

Saturation numbers for families of graph subdivisions

For a family of graphs, a graph G is ‐saturated if G contains no member of as a subgraph, but for any edge in , contains some member of as a subgraph. The minimum number of edges in an ‐saturated graph of order n is denoted . A subdivision of a graph H, or an H‐subdivision, is a graph G obtained from H by replacing the edges of H with internally disjoint paths of arbitrary length. We let denote the family of H‐subdivisions, including H itself. In this paper, we study when H is one of or , obtaining several exact results and bounds. In particular, we determine exactly for and show for n sufficiently large that there exists a constant such that . For we show that will suffice, and that this can be improved slightly depending on the value of . We also give an upper bound on for all t and show that . This provides an interesting contrast to a 1937 result of Wagner (Math Ann, 114 (1937), 570–590), who showed that edge‐maximal graphs without a K₅‐minor have at least edges.     

Stability of Hereditary Graph Classes Under Closure Operations

If is a subclass of the class of claw‐free graphs, then is said to be stable if, for any , the local completion of G at any vertex is also in . If is a closure operation that turns a claw‐free graph into a line graph by a series of local completions and is stable, then for any . In this article, we study stability of hereditary classes of claw‐free graphs defined in terms of a family of connected closed forbidden subgraphs. We characterize line graph preimages of graphs in families that yield stable classes, we identify minimal families that yield stable classes in the finite case, and we also give a general background for techniques for handling unstable classes by proving that their closure may be included into another (possibly stable) class.     

On Choosability with Separation of Planar Graphs with Forbidden Cycles

We study choosability with separation which is a constrained version of list coloring of graphs. A ‐list assignment L of a graph G is a function that assigns to each vertex v a list of at least k colors and for any adjacent pair , the lists and share at most d colors. A graph G is ‐choosable if there exists an L‐coloring of G for every ‐list assignment L. This concept is also known as choosability with separation. We prove that planar graphs without 4‐cycles are (3, 1)‐choosable and that planar graphs without 5‐ and 6‐cycles are (3, 1)‐choosable. In addition, we give an alternative and slightly stronger proof that triangle‐free planar graphs are (3, 1)‐choosable.     

On the Circular Chromatic Number of Graph Powers

This article intends to study some functors from the category of graphs to itself such that, for any graph G, the circular chromatic number of is determined by that of G. In this regard, we investigate some coloring properties of graph powers. We show that provided that . As a consequence, we show that if , then . In particular, and has no subgraph with circular chromatic number equal to . This provides a negative answer to a question asked in (X. Zhu, Discrete Math, 229(1–3) (2001), 371–410). Moreover, we investigate the nth multichromatic number of subdivision graphs. Also, we present an upper bound for the fractional chromatic number of subdivision graphs. Precisely, we show that .