Existence of 3‐chromatic Steiner quadruple systems

A Steiner quadruple system of order v (briefly SQS (v)) is a pair (X, ), where X is a v‐element set and is a set of 4‐element subsets of X (called blocks or quadruples), such that each 3‐element subset of X is contained in a unique block of . The chromatic number of an SQS(v)(X, ) is the smallest m for which there is a map such that for all , where . The system (X, ) is equitably m‐chromatic if there is a proper coloring with minimal m for which the numbers differ from each other by at most 1. Linek and Mendelsohn showed that an equitably 3‐chromatic SQS(v) exists for v ≡ 4, 8, 10 (mod 12), v ≥ 16. In this article we show that an equitably 3‐chromatic SQS(v) exists for v ≡ 2 (mod 12) with v > 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 469–477, 2007     

Ore‐type degree conditions for a graph to be H‐linked

Given a fixed multigraph H with V(H) = {h₁,…, h_m}, we say that a graph G is H‐linked if for every choice of m vertices v₁, …, _vm in G, there exists a subdivision of H in G such that for every i, v_i is the branch vertex representing h_i. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008     

Vertex suppression in 3‐connected graphs

To suppress a vertex in a finite graph G means to delete it and add an edge from a to b if a, b are distinct nonadjacent vertices which formed the neighborhood of . Let be the graph obtained from by suppressing vertices of degree at most 2 as long as it is possible; this is proven to be well defined. Our main result states that every 3‐connected graph G has a vertex x such that is 3‐connected unless G is isomorphic to , , or to a wheel for some . This leads to a generator theorem for 3‐connected graphs in terms of series parallel extensions. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 41–54, 2008     

Circular chromatic index of Cartesian products of graphs

The circular chromatic index of a graph G, written , is the minimum r permitting a function such that whenever e and are incident. Let □ , where □ denotes Cartesian product and H is an ‐regular graph of odd order, with (thus, G is s‐regular). We prove that , where is the minimum, over all bases of the cycle space of H, of the maximum length of a cycle in the basis. When and m is large, the lower bound is sharp. In particular, if , then □ , independent of m. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 7–18, 2008     

Maximum number of colorings of (2k,k2)‐graphs

Let consist of all simple graphs on 2k vertices and edges. For a simple graph G and a positive integer , let denote the number of proper vertex colorings of G in at most colors, and let . We prove that and is the only extremal graph. We also prove that as . © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 135–148, 2007     

Graph classes characterized both by forbidden subgraphs and degree sequences

Given a set of graphs, a graph G is ‐free if G does not contain any member of as an induced subgraph. We say that is a degree‐sequence‐forcing set if, for each graph G in the class of ‐free graphs, every realization of the degree sequence of G is also in . We give a complete characterization of the degree‐sequence‐forcing sets when has cardinality at most two. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 131–148, 2008     

A S(3, {4, 6}, 18) with a subdesign S(3, 4, 8) does not exist

For , a S(t,K,v) design is a pair, , with |V| = v and a set of subsets of V such that each t‐subset of V is contained in a unique and for all . If , , , and is a S(t,K,u) design, then we say has a subdesign on U. We show that a S(3,{4,6},18) design with a subdesign S(3,4,8) does not exist. © 2007 Wiley Periodicals, Inc. J Combin Designs 17: 36–38, 2009     

Coupled choosability of plane graphs

A plane graph G is coupled k‐choosable if, for any list assignment L satisfying for every , there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors. We prove that every plane graph is coupled 7‐choosable. We further show that maximal plane graphs, ‐minor free graphs, and plane graphs with maximum degree at most three are coupled 6‐choosable. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 27–44, 2008     

Disjoint cycles with chords in graphs

Let be integers, , , and let for each , be a cycle or a tree on vertices. We prove that every graph G of order at least n with contains k vertex disjoint subgraphs , where , if is a tree, and is a cycle with chords incident with a common vertex, if is a cycle. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 87–98, 2009     

On k‐detour subgraphs of hypercubes

A spanning subgraph G of a graph H is a k‐detour subgraph of H if for each pair of vertices , the distance, , between x and y in G exceeds that in H by at most k. Such subgraphs sometimes also are called additive spanners. In this article, we study k‐detour subgraphs of the n‐dimensional cube, , with few edges or with moderate maximum degree. Let denote the minimum possible maximum degree of a k‐detour subgraph of . The main result is that for every and On the other hand, for each fixed even and large n, there exists a k‐detour subgraph of with average degree at most . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 55–64, 2008     

Ore‐type condition for the existence of connected factors

In this article, we obtain some Ore‐type sufficient conditions for a graph to have a connected factor with degree restrictions. Let α and k be positive integers with if and if . Let G be a connected graph with a spanning subgraph F, each component of which has order at least α. We show that if the degree sum of two nonadjacent vertices is greater than then G has a connected subgraph in which F is contained and every vertex has degree at most . From the result, we derive that a graph G has a connected ‐factor if the degree sum of two nonadjacent vertices is at least . © Wiley Periodicals, Inc. J. Graph Theory 56: 241–248, 2007     

On conjectures of Frankl and El‐Zahar

An induced subgraph of a graph is called a derived subgraph of if contains no isolated vertices. An edge e of is said to be residual if e occurs in more than half of the derived subgraphs of . In this article, we prove that every simple graph with at least one edge contains a non‐residual edge. This was conjectured by El‐Zahar in 1997. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 344–352, 2008     

A better bound for the cop number of general graphs

In this note, we prove that the cop number of any n‐vertex graph G, denoted by , is at most . Meyniel conjectured . It appears that the best previously known sublinear upper‐bound is due to Frankl, who proved . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 45–48, 2008     

Cutting two graphs simultaneously

Consider two graphs, and , on the same vertex set V, with and having edges for . We give a simple algorithm that partitions V into sets A and B such that and . We also show, using a probabilistic method, that if and belong to certain classes of graphs, (for instance, if and both have a density of at least 2/, or if and are both regular of degree at most with n sufficiently large) then we can find a partition of V into sets A and B such that for . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 19–32, 2008     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The cover time of random geometric graphs

We study the cover time of random geometric graphs. Let $I(d)=[0,1]^{d}$ denote the unit torus in d dimensions. Let $D(x,r)$ denote the ball (disc) of radius r. Let $\Upsilon_d$ be the volume of the unit ball $D(0,1)$ in d dimensions. A random geometric graph $G=G(d,r,n)$ in d dimensions is defined as follows: Sample n points V independently and uniformly at random from $I(d)$ . For each point x draw a ball $D(x,r)$ of radius r about x. The vertex set $V(G)=V$ and the edge set $E(G)=\{\{v,w\}: w\ne v,\,w\in D(v,r)\}$ . Let $G(d,r,n),\,d\geq 3$ be a random geometric graph. Let $C_G$ denote the cover time of a simple random walk on G. Let $c>1$ be constant, and let $r=(c\log n/(\Upsilon_dn))^{1/d}$ . Then whp the cover time satisfies © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 324–349, 2011     

A system of equations for describing cocyclic Hadamard matrices

Given a basis for 2‐cocycles over a group G of order , we describe a nonlinear system of 4t‐1 equations and k indeterminates over , whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that ( ) is a solution of the system if and only if the 2‐cocycle gives rise to a cocyclic Hadamard matrix . Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups and . A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups , we have found that it suffices to check t instead of the 4t rows of , to decide the Hadamard character of the matrix (for a special class of cocycles f). © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 276–290, 2008     

On the existence of a rainbow 1‐factor in 1‐factorizations of K

Let be a 1‐factorization of the complete uniform hypergraph with and . We show that there exists a 1‐factor of whose edges belong to n different 1‐factors in . Such a 1‐factor is called a “rainbow” 1‐factor or an “orthogonal” 1‐factor. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 487–490, 2007     

Density of universal classes of series‐parallel graphs

A class of graphs ordered by the homomorphism relation is universal if every countable partial order can be embedded in . It is known (see [ 1 , 3 ]) that the class of k‐colorable graphs, for any fixed , induces a universal partial order. In 4 , a surprisingly small subclass of which is a proper subclass of the series‐parallel graphs (the K₄‐minor‐free graphs) is shown to be universal. On another side, Pan and Zhu in 7 proved a density result that for each rational number , there is a K₄‐minor‐free graph with circular chromatic number equal to a/b. In this note, we show for each rational number a/b within this interval the class of K₄‐minor‐free graphs with circular chromatic number a/b is universal if and only if , 5/2 or 3. This shows yet another surprising richness of the K₄‐minor‐free class that it contains universal classes as dense as the rational numbers. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Upper chromatic number of finite projective planes

For a finite projective plane , let denote the maximum number of classes in a partition of the point set, such that each line has at least two points in the same partition class. We prove that the best possible general estimate in terms of the order of projective planes is , which is tight apart from a multiplicative constant in the third term :

• (1) As holds for every projective plane of order q.
• (2) If q is a square, then the Galois plane of order q satisfies .

Our results asymptotically solve a ten‐year‐old open problem in the coloring theory of mixed hypergraphs, where is termed the upper chromatic number of . Further improvements on the upper bound (1) are presented for Galois planes and their subclasses. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 221–230, 2008