Anthony Hilton

• University of Reading

We show that for a simple graph $G$, $c'(G)\leq\Delta(G)+2$ where $c'(G)$ is the choice index (or edge-list chromatic number) of $G$, and $\Delta(G)$ is the maximum degree of $G$. As a simple corollary of this result, we show that the total chromatic number $\chi_T(G)$ of a simple graph satisfies the inequality $\chi_T(G)\leq\ \Delta(G)+4$ and the...

For d≥1,s≥0, a (d,d+s)-graph is a graph whose degrees all lie in the interval {d,d+1,…,d+s}. For r≥1,a≥0, an (r,r+a)-factor of a graph G is a spanning (r,r+a)-subgraph of G. An (r,r+a)-factorization of a graph G is a decomposition of G into edge-disjoint (r,r+a)-factors. A pseudograph is a graph which may have multiple edges and may have multiple l...

We generalize a theorem of M. Hall Jr., that an Latin rectangle on n symbols can be extended to an Latin square on the same n symbols. Let p, n, be positive integers such that and . Call an matrix on n symbols an -latinized rectangle if no symbol occurs more than once in any row or column, and if the symbol occurs at most times altogether . We give...

A function between graphs is k-to-1 if each point in the codomain has precisely k preimages in the domain. In this article, we approach the topic of continuous, or finitely discontinuous, k-to-1 functions between graphs from three different points of view. Harrold (Duke Math J 5 (1939), 789–793) showed that there is no 2-to-1 continuous function fr...

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can b...

Hall's Condition is a necessary condition for a partial latin square to be completable. Hilton and Johnson showed that for a partial latin square whose filled cells form a rectangle, Hall's Condition is equivalent to Ryser's Condition, which is a necessary and sufficient condition for completability. We give what could be regarded as an extension o...

In 1974 Cruse gave necessary and sufficient conditions for an r × s partial latin square P on symbols σ1,σ2,…,σt, which may have some unfilled cells, to be completable to an n × n latin square on symbols σ1,σ2,…,σn, subject to the condition that the unfilled cells of P must be filled with symbols chosen from {σt + 1,σt + 2,…,σn}. These conditions c...

Hall's condition is a well-known necessary condition for the existence of a proper coloring of a graph from prescribed lists. Completing a partial latin square is a very special kind of graph list-coloring problem. Cropper's question was: is Hall's condition sufficient for the existence of a completion of a partial latin square? The folk belief tha...

A graph G is r-starred if, for some v ∈ V(G), a largest pairwise intersecting family of independent r-subsets of V(G) may be obtained by taking all such subsets containing v (the r-star at v). Let G be the disjoint union of powers of cycles; Hilton and Spencer have studied the problem of determining the values of r for which G is r-starred. They co...

An (r,r+1)-factor of a graph G is a spanning subgraph H such that dH(v){r,r+1} for all vertices v(G). If G is expressed as the union of edge-disjoint (r,r+1)-factors, then this expression is an (r,r+1)-factorization of G. Let μ(r) be the smallest integer with the property that if G is a regular loopless multigraph of degree d with d≥μ(r), then G ha...

For d >= 1, s >= 0 a (d, d + s)-graph is a graph whose degrees all lie in the interval {d, d + 1, ..., d + s}. For r >= 1, a >= 0 an (r, r + a)-factor of a graph G is a spanning (r, r + a)-subgraph of G. An (r, r + a)-factorization of a graph G is a decomposition of G into edge-disjoint (r, r + a)-factors. A graph is (r, r + a)-factorable if it has...

A function between graphs is k-to-1 if each point in the co-domain has precisely k pre-images in the domain. Given two graphs, G and H, and an integer k≥1, and considering G and H as subsets of R3, there may or may not be a k-to-1 continuous function (i.e. a k-to-1 map in the usual topological sense) from G onto H. In this paper we review and compl...

We define the Hall strength s(M) of a partial Latin square M to be the least number of Hall inequalities whose satisfaction ensures that any partial Latin square of the same shape as M can be completed, and we let s(M)=∞ if there is no such least number. We give bounds for s(M) in a number of cases. We also define the conjugate Hall strength s c (M...

We show that the four-cycle has a k-fold list coloring if the lists of colors available at the vertices satisfy the necessary Hall's condition, and if each list has length at least ⌈5k/3⌉; furthermore, the same is not true with shorter list lengths. In terms of h(k)(G), the k -fold Hall number of a graph G, this result is stated as h(k)(C4)=2k−⌊k/3...

When trying to construct a school timetable, a good first step might seem to be to construct an outline timetable in which all History teachers are counted together, all French teachers are counted together, etc., all classes of each year group are counted together, and in which the preliminary division is into days rather than lessons. Having cons...

A well known conjecture in graph theory states that every regular graph of even order and degree , where , is 1-factorizable. Chetwynd and Hilton [A.G. Chetwynd, A.J.W. Hilton, 1-factorizing regular graphs of high degree — An improved bound, Discrete Math. 75 (1989) 103-112] and, independently, Niessen and Volkmann [T. Niessen, L. Volkmann, Class 1...

Let G be a graph consisting of powers of disjoint cycles and let AA be an intersecting family of independent r-sets of vertices. Provided that G satisfies a further condition related to the clique numbers of the powers of the cycles, then |A||A| will be as large as possible if it consists of all independent r-sets containing one vertex from a speci...

For integers d≥0, s≥0, a (d, d+s)-graph is a graph in which the degrees of all the vertices lie in the set {d, d+1, …, d+s}. For an integer r≥0, an (r, r+1)-factor of a graph G is a spanning (r, r+1)-subgraph of G. An (r, r+1)-factorization of a graph G is the expression of G as the edge-disjoint union of (r, r+1)-factors. For integers r, s≥0, t≥1,...

A function between graphs is k-to-1 if each point in the codomain has precisely k pre-images in the domain. Given two graphs, G and H, and an integer k≥1, and considering G and H as subsets of ℝ3, there may or may not be a k-to-1 continuous function (i.e. a k-to-1 map in the usual topological sense) from G onto H. In this paper we consider gra...

A (d,d+1)(d,d+1)-graph is a graph whose vertices all have degrees in the set {d,d+1}{d,d+1}. Such a graph is semiregular. An (r,r+1)(r,r+1)-factorization of a graph G is a decomposition of G into (r,r+1)(r,r+1)-factors. For d-regular simple graphs G we say for which x and r G must have an (r,r+1)(r,r+1)-factorization with exactly x(r,r+1)(r,r+1)-fa...

We prove that any cycle

In an ordinary list multicoloring of a graph, the vertices are “colored” with subsets of pre-assigned finite sets (called “lists”) in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the col...

A family A of r-subsets of the vertex set V(G) of a graph G is intersecting if any two of the r-subsets have a non-empty intersection. The graph G is r-EKR if a largest intersecting family A of independent r-subsets of V(G) may be obtained by taking all independent r-subsets containing some particular vertex. In this paper, we show that if G consis...

We improve an upper bound for the chromatic index of a multigraph due to Andersen and Gol'dberg. As a corollary we deduce that if no two edges of multiplicity at least two in G are adjacent, then χ′(G) ⩽ Δ(G) + 1. In addition we generalize results concerning the structure of critical graphs due to Vizing and to Chetwynd and Hilton.

Let K 2n+1(r) denote the complete graph K2n+1 with each edge replicated r times and let χ′(G) denote the chromatic index of a multigraph G. A multigraph G is critical if χ′(G) > χ′(G/e) for each edge e of G. Let S be a set of sn – 1 edges of K 2n+1(r). We show that, for 0 < s ≦ r, G/S is critical and that χ′ (G/(S ∪{e})) = 2rn + r – s for all e ∈ E...

We show that, for r = 1, 2, a graph G with 2n + 2 (≥6) vertices and maximum degree 2n + 1 - r is of Class 2 if and only if |E(G/v)| > () - rn, where v is a vertex of G of minimum degree, and we make a conjecture for 1 ≤ r ≤ n, of which this result is a special case. For r = 1 this result is due to Plantholt.

For a positive integer n, let G be Kn if n is odd and Kn less a one-factor if n is even. In this paper it is shown that, for non-negative integers p, q and r, there is a decomposition of G into p 4-cycles, q 6-cycles and r 8-cycles if 4p + 6q + 8r = |E(G)|, q = 0 if n < 6, and r = 0 if n < 8.

In this paper, we give two sufficient conditions for a graph to be type 1 with respect to the total chromatic number and prove the following results. (i) if G and H are of type 1, then G × H is of type 1; (ii) if ε(G) ≤ v(G) + | 3/2Δ(G) - 4, then G is of type 1.

A finite latin square is an n×n matrix whose entries are elements of the set {1,…,n} and no element is repeated in any row or column. Given equivalence relations on the set of rows, the set of columns, and the set of symbols, respectively, we can use these relations to identify equivalent rows, columns and symbols, and obtain an amalgamated latin s...

Let ∞(G) denote the domination number of a graph, and let C be the set of all Hamiltonian cubic graphs. Let

Those (2m - 1)-edge-colourings of a spanning subgraph of K2m, consisting of Kr and independent edges, that can be embedded in a (2m - l)-edge-colouring of K2m are characterised.

A graph G is a (d, d + s)-graph if the degree of each vertex of G lies in the interval [d, d + s]. A (d, d + 1)-graph is said to be semiregular. An (r, r + 1) -factorization of a graph is a decomposition of the graph into edgedisjoint (r, r + 1)-factors.We discuss here the state of knowledge about (r, r + 1)-factorizations of d -regular graphs and...

Let t be a positive integer, and let L=(l1,…,lt) and K=(k1,…,kt) be collections of nonnegative integers. A graph has a (t,K,L) factorization if it can be represented as the edge-disjoint union of factors F1,…,Ft where, for 1⩽i⩽t, Fi is ki-regular and at least li-edge-connected. In this paper we consider (t,K,L)-factorizations of complete equipartit...

Let t be a positive integer, and let L = (l 1 , . . . , l t ) and K = (k 1 , . . . , k t ) be collections of nonnegative integers. A graph has a (t, K,L) factorization if it can be represented as the edge-disjoint union of factors F 1 , . . . , F t where, for 1 and at least l i -edge-connected. In this paper we consider (t, K,L)- factorizations of...

We show how to find a decomposition of the edge set of the complete graph into regular factors where the degree and edge-connectivity of each factor is given.

We show that, in the special case when S=(2,1,⋯,1), an S-outline transitive triple system is an S-amalgamated transitive triple system.

In this paper, necessary and sufficient conditions are found for a graph with exactly one amalgamated vertex to be the amalgamation of a k-factorization of Kkn+1 in which each k-factor is connected. From this result, necessary and sufficient conditions for a given edge-coloured Kt to be embedded in a connected k-factorization of Kkn+1 are deduced.

We show how to find a decomposition of the edge set of the complete graph into regular factors where the degree and edge-connectivity of each factor is prescribed. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 132–136, 2003

It has been known for some time that if the Conformability Conjecture is true, then we can say exactly which graphs G of maximum degree and even order are totally critical; by contrast, the situation for odd order graphs has been unclear. Here we look in particular at odd order graphs G with low deficiency and high maximum degree relative to the or...

A double latin square of order 2n on symbols σ1,…,σn is a 2n×2n matrix A=(aij) in which each aij is one of the symbols σ1,…,σn and each σk occurs twice in each row and twice in each column. For k=1,…,n let B(A,σk) be the bipartite graph with vertices ρ1,…,ρ2n,c1,…,c2n and 4n edges [ρi,cj] corresponding to ordered pairs (i,j) such that aij=σk. We sa...

Given a graph G, for each υ ∈V(G) let L(υ) be a list assignment to G. The well-known choice number c(G) is the least integer j such that if |L(υ)| ≥j for all υ ∈V(G), then G has a proper vertex colouring ϕ with ϕ(υ) ∈ L (υ) (∀υ ∈V(G)). The Hall number h(G) is like the choice number, except that an extra non-triviality condition, called Hall's condi...

We show that if G and H are non-conformable graphs, with H being a subgraph of G of the same maximum degree Δ(G), and if , then |V(H)|=|V(G)|. We also show that this inequality is best possible, for when there are examples of graphs G and H with Δ(H)=Δ(G) and |V(H)|<|V(G)| which are both non-conformable. We determine all such examples. Interest in...

A triangulated graph is an outline triple system of even index if and only if it is an amalgamated triple system of even index.

The well-known Oberwolfach problem is to show that it is possible to 2-factorize K n (n odd) or K n less a 1-factor (n even) into predetermined 2-factors, all isomorphic to each other; a few exceptional cases where it is not possible are known. A completely new technique is introduced that enables it to be shown that there is a solution when each 2...

In this paper we show that under some fairly general conditions the Overfull Conjecture about the chromatic index of a graph G implies the Conformability Conjecture about the total chromatic number of G. We also show that if G has even order and high maximum degree, then G is conformable unless the deficiency is very small.

Hall's condition for the existence of a proper vertex list-multicoloring of a simple graph G has recently been used to define the fractional Hall and Hall-condition numbers of G, hf(G) and sf(G). Little is known about hf(G), but it is known that sf(G)=max[|V(H)|/α(H);H⩽G], where ‘⩽’ means ‘is a subgraph of’ and α(H) denotes the vertex independence...

An assignment of colours to the edges of a multigraph is called an s-improper edge-colouring if no colour appears on more than s edges incident with any given vertex. We prove that if L:E(G) -> 2(N) is an assignment of lists of colours to the edges of a multigraph G with \L(e)\ greater than or equal to [max{d(u). d(v)}/s] for every edge e joining v...

We prove that if a triangulated digraph is an pseudo-outline transitive triple system then it is a pseudo-amalgamated transitive triple system.

Recently the authors defined the concept of a weighted quasigroup, and showed that each weighted quasigroup is the amalgamation of a quasigroup. Similar results were obtained for symmetric and other types of quasigroups.Here we first introduce the closely related concept of a fractional latin square and show that every fractional latin square is th...

Philip Hall's famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list-colorings or list-multicolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural...

The core of a graph G is the subgraph GΔ induced by the vertices of maximum degree. We define the deleted core D(G) of G. We extend an earlier sufficient condition due to Hoffman [7] for a graph H to be the core of a Class 2 graph, and thereby provide a stronger sufficient condition. The new sufficient condition is in terms of D(H). We show t...

If r|(n-1) and rn is even, then Kn can be expressed as the union of edge-disjoint isomorphic r-regular r-connected factors. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 15–21, 2000

The graphs with Hall number at most 2 form a class of graphs within which the chromatic number equals the choice (list-chromatic) number. This class has a forbidden-induced-subgraph characterization which has not yet been found, although a fairly imposing collection of minimal forbidden induced subgraphs has been assembled. In this paper we add to...

A graph is totally critical if it is Type 2, connected, and the removal of any edge reduces the total chromatic number. A good characterization of all totally critical graphs is unlikely as Sanchez-Arroyo showed that determining the total chromatic number of a graph is an NP-hard problem. In this paper we show that if the Conformability Conjecture...

We discuss the relationship between the Hall number of a graph and other well-known graph parameters, including the choice number and the chromatic number. We also discuss the edge and total analogues of these relationships.

We prove a new vertex-splitting lemma which states that if a multigraphGhas maximum multiplicity of at mostp, then each vertex u can be split into ⌈(d(u)/p)⌉ new vertices, ⌊(d(u)/p)⌋ of degreep, with the multiple edges being shared out between the new vertices in such a way that each multiple edge remains intact at at least one of its two endpoints...

If G is a graph of order $2n \geq 4$ with an equibipartite complement, then G is Class 1 (i.e., the chromatic index of G is Δ (G)) if and only if G is not the union of two disjoint Kn's with n odd. Similarly if G is a graph of order 2n ≥ 6 whose complement G is equibipartite with bipartition (A, D), and if both G and B, the induced bipartite subgra...

We study graphs which are critical with respect to the chromatic index. We relate these to the Overfull Conjecture and we study in particular their construction from regular graphs by subdividing an edge or by splitting a vertex.

If G is a regular tripartite graph of degree d(G) with tripartition (A,B,C) of V(G) such that the bipartite subgraphs induced by each of A ∪ B, B ∪ C, C ∪ A are all regular of degree , then we call G 3-way regular. We give necessary and sufficient conditions for a 3-way regular tripartite graph of degree 4 to have a decomposition into edge-disjoint...

Suppose that f is a (⩽k)-to-1 function from a vertex set of a graph G onto a vertex set of a graph H. We ask when f extends to a continuous (⩽k)-to-1 map from G onto H. In an earlier paper, the authors answered this question, for the case that k is odd, with local conditions only on the adjacency matrix for H and the inverse adjacency matrix for G....

We give a complete and explicit characterization of the connected graphs which admit a continuous 3-to-l map onto the circle, and of the connected graphs which admit a continuous 2-to-1 map onto the circle. This generalizes earlier work of Heath and Hilton who considered the mappings of trees onto the circle.

A proper coloring of a finite simple graph G from an assignment of lists (sets) to the vertices of G can also be regarded as a “system of G-distinct representatives” of the sets on the vertices. We consider a crude necessary condition for such a system that we call Hall’s condition because of a strong family resemblance to the condition for the exi...

It is shown that the counterexamples to the conformability conjecture found by Chen and Fu possess a property which is (essentially) not possessed by any other graph satisfying the conditions of the conjecture. Furthermore it is shown that it is this property which is fundamental to the Chen and Fu graphs being counterexamples to the conjecture.

We give some sufficient conditions for an (S, U)-outline T-factorization of Kn to be an (S, U)-amalgamated T-factorization of Kn. We then apply these to give various necessary and sufficient conditions for edge coloured graphs G to have recoverable embeddings in T-factorized Kn's.

An edge-colouring of a graph G is equitable if, for each vertex v of G, the number of edges of any one colour incident with v differs from the number of edges of any other colour incident with v by at most one. We show that if k⩾2 and k∤d(v) (∀vϵV(G)) and G is a simple graph, then G has an equitable edge-colouring with k colours. This result is als...

We show that a regular graph G of order at least 6 whose complement G is bipartite has total chromatic number d(G)+1 if and only if (i) G is not a complete graph, and (ii) G not equal K-n,K-n($) over bar when n is even. As an aid in the proof of this, we also show that, for n greater than or equal to 4, if the edges of a Hamiltonian path of K-2n ar...

With the proof of the Evans conjecture, it was established that any partial latin square of side n with a most n − 1 nonempty cells can be completed to a latin square of side n. In this article we prove an analogous result for symmetric latin squares: a partial symmetric latin square of side n with an admissible diagonal and at most n − 1 nonempty...

The core GΔ of a simple graph G is the subgraph induced by the vertices of maximum degree. It is well known that the Petersen graph is not 1-factorizable and has property that the core of the graph obtained from it by removing one vertex has maximum degree 2. In this paper, we prove the following result. Let G be a regular graph of even order with...

Jo Heath and the author found a set of necessary and sufficient conditions for the existence of an exactly k-to-1 map from a graph G to a graph H. These conditions were all local ones. Here this result is used to give necessary and sufficient conditions for the existence of an exactly k-to-1 map from G to H when k is sufficiently large. The conditi...

This volume comprises the invited lectures given at the 14th British Combinatorial Conference. The lectures survey many topical areas of current research activity in combinatorics and its applications, and also provide a valuable overview of the subject, for both mathematicians and computer scientists.

The total colouring conjecture is shown to be correct for those graphs G having .

We give a survey of various recent results concerning the total chromatic number of simple graphs.

The Δ-subgraph G Δ of a simple graph G is the subgraph induced by the vertices of maximum degree of G. In this paper, we obtain some results about the construction of a graph G if the graph G is Class 2 and the structure of G Δ is particularly simple.

We give necessary and sufficient conditions for the existence of an alternating Hamiltonian cycle in a complete bipartite graph whose edge set is colored with two colors.

Suppose k is a positive integer, G and H are graphs, and f is a k-to-1 correspondence from a vertex set of G onto a vertex set of H. Conditions on the adjacency matrices are given that are necessary and sufficient for f to extend to a continuous k-to-1 map from G onto H.