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.     

Average Degree in Graph Powers

The kth power of a simple graph G, denoted by , is the graph with vertex set where two vertices are adjacent if they are within distance k in G. We are interested in finding lower bounds on the average degree of . Here we prove that if G is connected with minimum degree and , then G⁴ has average degree at least . We also prove that if G is a connected d‐regular graph on n vertices with diameter at least , then the average degree of is at least Both these results are shown to be essentially best possible; the second is best possible even when is arbitrarily large.     

Improper Coloring of Sparse Graphs with a Given Girth,II: Constructions

A graph G is ‐colorable if can be partitioned into two sets and so that the maximum degree of is at most j and of is at most k. While the problem of verifying whether a graph is (0, 0)‐colorable is easy, the similar problem with in place of (0, 0) is NP‐complete for all nonnegative j and k with . Let denote the supremum of all x such that for some constant every graph G with girth g and for every is ‐colorable. It was proved recently that . In a companion paper, we find the exact value . In this article, we show that increasing g from 5 further on does not increase much. Our constructions show that for every g, . We also find exact values of for all g and all .     

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 .     

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 .     

On Rooted Packings,Decompositions, and Factors of Graphs

In this article, we study so‐called rooted packings of rooted graphs. This concept is a mutual generalization of the concepts of a vertex packing and an edge packing of a graph. A rooted graph is a pair , where G is a graph and . Two rooted graphs and are isomorphic if there is an isomorphism of the graphs G and H such that S is the image of T in this isomorphism. A rooted graph is a rooted subgraph of a rooted graph if H is a subgraph of G and . By a rooted ‐packing into a rooted graph we mean a collection of rooted subgraphs of isomorphic to such that the sets of edges are pairwise disjoint and the sets are pairwise disjoint. In this article, we concentrate on studying maximum ‐packings when H is a star. We give a complete classification with respect to the computational complexity status of the problems of finding a maximum ‐packing of a rooted graph when H is a star. The most interesting polynomial case is the case when H is the 2‐edge star and S contains the center of the star only. We prove a min–max theorem for ‐packings in this case.     

Face 2‐Colorable Embeddings with Faces of Specified Lengths

Suppose and are arbitrary lists of positive integers. In this article, we determine necessary and sufficient conditions on M and N for the existence of a simple graph G, which admits a face 2‐colorable planar embedding in which the faces of one color have boundary lengths and the faces of the other color have boundary lengths . Such a graph is said to have a planar ‐biembedding. We also determine necessary and sufficient conditions on M and N for the existence of a simple graph G whose edge set can be partitioned into r cycles of lengths and also into t cycles of lengths . Such a graph is said to be ‐decomposable.     

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.     

Toughness and Vertex Degrees

We study theorems giving sufficient conditions on the vertex degrees of a graph G to guarantee G is t‐tough. We first give a best monotone theorem when , but then show that for any integer , a best monotone theorem for requires at least nonredundant conditions, where grows superpolynomially as . When , we give an additional, simple theorem for G to be t‐tough, in terms of its vertex degrees.     

Minors in ‐ Chromatic Graphs,II

Let denote the graph obtained from the complete graph by deleting the edges of some ‐subgraph. The author proved earlier that for each fixed s and , every graph with chromatic number has a minor. This confirmed a partial case of the corresponding conjecture by Woodall and Seymour. In this paper, we show that the statement holds already for much smaller t, namely, for .     

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.     

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 .     

On the Multichromatic Number of s‐Stable Kneser Graphs

For positive integers n and s, a subset [n] is s‐stable if for distinct . The s‐stable r‐uniform Kneser hypergraph is the r‐uniform hypergraph that has the collection of all s‐stable k‐element subsets of [n] as vertex set and whose edges are formed by the r‐tuples of disjoint s‐stable k‐element subsets of [n]. Meunier ( 21 ) conjectured that for positive integers with , and , the chromatic number of s‐stable r ‐uniform Kneser hypergraphs is equal to . It is a generalized version of the conjecture proposed by Alon et al. ( 1 ). Alon et al. ( 1 ) confirmed Meunier's conjecture for with arbitrary positive integer q. Lin et al. ( 17 ) studied the kth chromatic number of the Mycielskian of the ordinary Kneser graphs for . They conjectured that for . The case was proved by Mycielski ( 22 ). Lin et al. ( 17 ) confirmed their conjecture for , or when n is a multiple of k or . In this paper, we investigate the multichromatic number of the usual s ‐stable Kneser graphs . With the help of Fan's (1952) combinatorial lemma, we show that Meunier's conjecture is true for r is a power of 2 and s is a multiple of r, and Lin‐Liu‐Zhu's conjecture is true for .     

Multicolor Ramsey Numbers For Complete Bipartite Versus Complete Graphs

Let be graphs. The multicolor Ramsey number is the minimum integer r such that in every edge‐coloring of by k colors, there is a monochromatic copy of in color i for some . In this paper, we investigate the multicolor Ramsey number , determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different constructions are used for the lower bounds, including the random graph and explicit graphs built from finite fields. A technique of Alon and Rödl using the probabilistic method and spectral arguments is employed to supply tight lower bounds. A sample result is for any t and m, where c₁ and c₂ are absolute constants.     

Pebbling Graphs of Fixed Diameter

Given a configuration of indistinguishable pebbles on the vertices of a connected graph G on n vertices, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one pebble on an adjacent vertex. The m‐pebbling number of a graph G, , is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least m pebbles on v. When , it is simply called the pebbling number of a graph. We prove that if G is a graph of diameter d and are integers, then , where denotes the size of the smallest distance k dominating set, that is the smallest subset of vertices such that every vertex is at most distance k from the set, and, . This generalizes the work of Chan and Godbole (Discrete Math 208 (2008), 15–23) who proved this formula for . As a corollary, we prove that . Furthermore, we prove that if d is odd, then , which in the case of answers for odd d, up to a constant additive factor, a question of Bukh (J Graph Theory 52 (2006), 353–357) about the best possible bound on the pebbling number of a graph with respect to its diameter.     

The Images of the Clique Operator and Its Square are Different

Let be the class of all graphs and K be the clique operator. The validity of the equality has been an open question for several years. A graph in but not in is exhibited here.     

A Hypergraph Version of a Graph Packing Theorem by Bollobás and Eldridge

Two n‐vertex hypergraphs G and H pack, if there is a bijection such that for every edge , the set is not an edge in H. Extending a theorem by Bollobás and Eldridge on graph packing to hypergraphs, we show that if and n‐vertex hypergraphs G and H with with no edges of size 0, 1, and n do not pack, then either

1. one of G and H contains a spanning graph‐star, and each vertex of the other is contained in a graph edge, or
2. one of G and H has edges of size not containing a given vertex, and for every vertex x of the other hypergraph some edge of size does not contain x.

     

k‐Kings in k‐Quasitransitive Digraphs

Let D be a digraph with vertex set and arc set . A vertex x is a k‐king of D, if for every , there is an ‐path of length at most k. A subset N of is k‐independent if for every pair of vertices , we have and ; it is l‐absorbent if for every there exists such that . A ‐kernel of D is a k‐independent and l‐absorbent subset of . A k‐kernel is a ‐kernel. A digraph D is k‐quasitransitive, if for any path of length k, x₀ and are adjacent. In this article, we will prove that a k‐quasitransitive digraph with has a k‐king if and only if it has a unique initial strong component and the unique initial strong component is not isomorphic to an extended ‐cycle where each has at least two vertices. Using this fact, we show that every strong k‐quasitransitive digraph has a ‐kernel.     

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. ).