Complete Graph

Recent proposals for semantics of default theories are all based on some types of weaker notion of extensions This is typi ed in the well founded semantics and the extension class semantics for default theories Although these semantics solve the no extension problem in Reiter s default logic they also present a departure from Reiter s original extension semantics even for default theories which can be completely char acterized by the extension semantics This results in a weaker capability for skeptical reasoning In this paper we propose a semantics for default theories based on van Gelder s alternating xpoint theory The distinct feature of this semantics is its preservation of Reiter s semantics when a default theory is not considered problematic under Reiter s semantics Di erences arise only when a default theory has no extensions or has only biased extensions under Reiter s semantics This feature allows skeptical reasoning in Reiter s logic to be properly preserved in the new sema...

This paper proposes a formalism for nonmonotonic reasoning based on prioritized argumentation. We argue that nonmonotonic reasoning in general can be viewed as selecting monotonic inferences by a simple notion of priority among inference rules. More importantly, these types of constrained inferences can be speci ed in a knowledge representation language where a theory consists of a collection of rules of rst order formulas and a priority among these rules. We recast default reasoning as a form of prioritized argumentation, and illustrate how the parameterized formulation of priority may be used to allow various extensions and modi cations to default reasoning. We also show that it is possible, but more di cult, to express prioritized argumentation by default logic: even some particular forms of prioritized argumentation cannot be represented modularly by defaults under the same language.

Let R m be the (unique) universal homogeneous m-edge-coloured countable complete graph (m ≥ 2), and G m its group of colourpreserving automorphisms. The group G m was shown to be simple by John Truss. We examine the automorphism group of G m , and show that it is the group of permutations of R m which induce permutations on the colours, and hence an extension of G m by the symmetric group of degree m. We show further that the extension splits if and only if m is odd, and in the case where m is even and not divisible by 8 we find the smallest supplement for G m in its automorphism group.

The operation of switching a finite graph was introduced by Seidel, in the study of strongly regular graphs. We may conveniently regard a graph as being a 2-colouring of a complete graph; then the extension to switching of an m-coloured complete graph is easy to define. However, the situation is very different. For m > 2, all m-coloured graphs lie in the same switching class. However, there are still interesting things to say, especially in the infinite case. This paper presents the basic theory of switching with more than two colours. In the finite case, all graphs on a given set of vertices are equivalent under switching, and we determine the structure of the switching group and show that its extension by the symmetric group on the vertex set is primitive. In the infinite case, there is more than one switching class; we determine all those for which the group of switching automorphisms is the symmetric group. We also exhibit some other cases (including the random mcoloured complete graph) where the group of switching-automorphisms is highly transitive. Finally we consider briefly the case where not all switchings are allowed. For convenience, we suppose that there are three colours of which two may be switched. We show that, in the case of almost all finite random graphs, the analogue of the bijection between switching classes and two-graphs holds.

The operation of switching a finite graph was introduced by Seidel, in the study of strongly regular graphs. We may conveniently regard a graph as being a 2-colouring of a complete graph; then the extension to switching of an m-coloured complete graph is easy to define. However, the situation is very different. For m > 2, all m-coloured graphs lie in the same switching class. However, there are still interesting things to say, especially in the infinite case. This paper presents the basic theory of switching with more than two colours. In the finite case, all graphs on a given set of vertices are equivalent under switching, and we determine the structure of the switching group and show that its extension by the symmetric group on the vertex set is primitive. In the infinite case, there is more than one switching class; we determine all those for which the group of switching automorphisms is the symmetric group. We also exhibit some other cases (including the random mcoloured complete graph) where the group of switching-automorphisms is highly transitive. Finally we consider briefly the case where not all switchings are allowed. For convenience, we suppose that there are three colours of which two may be switched. We show that, in the case of almost all finite random graphs, the analogue of the bijection between switching classes and two-graphs holds.

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connectedhomogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.

Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of two pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding if a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and that deciding if the pebbling number has a prescribed upper bound is Π P 2 -complete. On the other hand, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter two graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and many authors have studied which graphs are Class 0 and what graph properties guarantee it, with no characterization in sight. In this paper we investigate an important family of diameter three chordal graphs called split graphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide a formula for the pebbling number of a split graph, along with an algorithm for calculating it that runs in O(n β ) time, where β = 2ω/(ω + 1) ∼ = 1.41 and ω ∼ = 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.

Graph pebbling models the transportation of consumable resources. As two pebbles move across an edge, one reaches its destination while the other is consumed. The t-pebbling number is the smallest integer m so that any initially distributed supply of m pebbles can place t pebbles on any target vertex via pebbling moves. The 1-pebbling number of diameter two graphs is well-studied. Here we investigate the t-pebbling number of diameter two graphs under the lense of connectivity.

Well-known necessary conditions for the existence of a generalized Bhaskar Rao design, GBRD(v, 3, λ; G) with v ≥ 4 are: (i ) λ ≡ 0 (mod |G|), (ii ) λ(v -1) ≡ 0 (mod 2), (iii ) λv(v -1) ≡ 0 (mod 3), (iv ) if |G| ≡ 0 (mod 2) then λv(v -1) ≡ 0 (mod 8). In this paper we show that these conditions are sufficient whenever (i ) the group G has odd order or (ii ) the order is of the form 2q for q = 3 m or q an odd number which is not a multiple of 3.

Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing parity of one of the descent's numbers which was extended and generalized in several ways in the literature. In this paper, we shall fix a set partition of the natural numbers N, (N1, . . . , Ns), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = {1, . . . , k}. In particular, we refine and generalize some of the results by Burstein and Mansour.

We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.

We find analytically the complete set of eigenvalues and eigenvectors associated with Metropolis dynamics on a complete graph. As an application, we use this information to study a counter-intuitive relaxation phenomenon, called the Mpemba effect. This effect describes situations when upon performing a thermal quench, a system prepared in equilibrium at high temperatures relaxes faster to the bath temperature than a system prepared at a temperature closer to that of the bath. We show that Metropolis dynamics on a complete graph does not support weak nor strong Mpemba effect, however, when the graph is not complete, the effect is possible.

In drawings (two edges have at most one point in common, either a node or a crossing) of the complete graph K, in the Euclidean plane there occur at most 2n -2 edges without crossings. This was proved by G. Ringel in [l!. Here the minimal number of edges without crossings in drawings of K,, is determined, and for the existence of values between minimum and maximum is asked. He also remarks in [l], that he knows a special D(K,,) in which no edge without crossings occurs. Here we will show that drawings D(K,J of this kind only exist if n >, 8. Moreover we will determine h(n) for the remaining values of n. 299

The minimum covering energy of a simple graph G is the sum of the absolute values of the eigenvalues of its minimum covering adjacency matrix Ac(G). This study investigates the minimum covering energy of standard graphs including complete graph, star graph, crown graph, complete bipartite graph, and cocktail graph using SCILAB programming. The Minimum covering energy is a critical parameter in network optimization, representing the minimum energy required to cover all edges. In this paper, we have newly constructed five different SCILAB programs for minimum covering energy of simple graph of order n using SCILAB version 6.0.1. The algorithm is tested on standard graphs with varying sizes, and the results demonstrate the approach's effectiveness. The minimum covering energy of a graph increases with graph size.

In this article, we extend the classic Banach contraction principle to a complete metric space equipped with a partial order. We accomplish this by establishing some coincidence point for h-non decreasing self mapping satisfying certain rational type contractions in the framework of a metric space equipped with partial order. These contributions extend the existing literature on metric spaces and fixed point theory. Through illustrative examples, we showcase the practical applicability of our proposed notions and results. In addition, we present applications of our main results for a self mapping involving the integral type contractions.

A family C of sets has the Helly property if any subfamily C ′ whose elements are pairwise intersecting has non-empty intersection. Suppose that C is a non-empty family of subsets of a finite set V : the Helly number h(C) of C is the least positive integer n such that every n-wise intersecting subfamily of C has non-empty intersection. In this paper the Helly property of families of convex sets relative to two new graph convexities are studied. Let G be a (finite) connected graph and U a set of vertices of G. A connected subgraph with the fewest edges containing U is called a Steiner tree for U, and the collection of all vertices of G that belong to some Steiner tree for U is called the Steiner interval for U. A set S of vertices of G is g 3 -convex if it contains the Steiner interval for every 3-subset U of S. A subtree T of G that contains U is a minimal U-tree if every vertex of T that is not in U is a cut-vertex of the subgraph induced by V (T ). The set of all vertices that belong to some minimal U-tree is called the monophonic interval for U and a set S of vertices is m 3 -convex if it contains the monophonic interval of every 3-subset U of S. Those graphs are characterized for which the families of g 3 -convex (m 3 -convex) sets of size at least 3 have the Helly property. A graph obtained from a complete graph by deleting a matching is called a near-clique. The maximum order of a near-clique in a graph G is called the near-clique number of G. The near-clique number of a graph is a lower bound on the Helly number for both g 3 -convex families and m 3 -convex families. For m 3 -convex families equality holds. For g 3 -convex families equality holds for chordal graphs and for distance-hereditary graphs, but the bound can be arbitrarily bad in general, even when the near-clique number is 3.

As an analogous concept of a nowhere-zero flow for directed graphs, zero-sum flows and constant-sum flows are defined and studied in the literature. For an undirected graph, a zero-sum flow (constant-sum flow resp.) is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero (constant h resp.), and we call it a zero-sum k-flow (h-sum k-flow resp.) if the values of the edges are less than k. We extend these concepts to general constant-sum A-flow, where A is an Abelian group, and consider the case A = Z k the additive Abelian cyclic group of integer congruences modulo k with identity 0. In the literature, a graph is alternatively called Z k -magic if it admits a constant-sum Z k -flow, where the constant sum is called a magic sum or an index for short. We define the set of all possible magic sums such that G admits a constant-sum Z k -flow to be I k (G) and call it the magic sum spectrum, or for short, the index set of G with respect to Z k . In this article, we study the general properties of the magic sum spectrum of graphs. We determine the magic sum spectrum of complete bipartite graphs K m,n for m ≥ n ≥ 2 as the additive cyclic subgroups of Z k generated by k d , where d = gcd(mn, k). Also, we show that every regular graph G with a perfect matching has a full magic sum spectrum, namely, I k (G) = Z k for all k ≥ 3. We characterize a 3-regular graph so that it admits a perfect matching if and only if it has a full magic sum spectrum, while an example is given for a 3-regular graph without a perfect matching which has no full magic sum spectrum. Another example is given for a 5-regular graph without a perfect matching, which, however, has a full magic sum spectrum. In particular, we completely determine the magic sum spectra for all regular graphs of even degree. As a byproduct, we verify a conjecture raised by Akbari et al., which claims that every connected 4k-regular graph of even order admits a 1-sum 4-flow. More open problems are included.

We consider multiuser communication on a binary input additive white Gaussian noise channel using Randomly Spread-Code Division Multiple Access. We show concentration of various quantities of the system including the capacity and the free energy. We also obtain a tight upper bound on the capacity of this system in the large user limit which matches with the replica solution. The method is quite general and can be extended to many other multiuser scenarios.

We show that the Hamming graph H(3, q) with diameter three is uniquely determined by its spectrum for q ≥ 36. Moreover, we show that for given integer D ≥ 2, any graph cospectral with the Hamming graph H(D, q) is locally the disjoint union of D copies of the complete graph of size q -1, for q large enough.

Extended logic programs and annotated logic programs are two important extensions of normal logic programs that allow for a more concise and declarative representation of knowledge. Extended logic programs add explicit negation to the default negation of normal programs in order to distinguish what can be shown to be false from what cannot be proven true. Annotated logic programs generalize the set of truth values over which a program is interpreted by explicitly annotating atoms with elements of a new domain of truth values. In this paper coherent well-founded annotated programs are de ned, and shown to generalize both consistent and paraconsistent extended programs, along with several classes of annotated programs.

A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The Diameter-Constrained MST (DCMST) problem can be stated as follows: given an undirected, edge-weighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NP-complete, for all values of k; 4 k n − 2). In this paper, we investigate the behavior of the diameter of MST in randomly-weighted complete graphs (in Erdös-Rényi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is small-independent of n, we present a one-time-tree-construction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of these heuristics on sequential and parallel machines. Our extensive empirical study shows that these two heuristics produce good solutions for a wide variety of inputs.

Isometric subgraphs of hypercubes, or partial cubes as they are also called, are a rich class of graphs that include median graphs, subdivision graphs of complete graphs, and classes of graphs arising in mathematical chemistry and biology. In general, one can recognize whether a graph on n vertices and m edges is a partial cube in O(mn) steps, faster recognition algorithms are only known for median graphs. This paper exhibits classes of partial cubes that are not median graphs but can be recognized in O(m log n) steps. On the way relevant decomposition theorems for partial cubes are derived, one of them correcting an error in a previous paper [W.

The P → ∞ limit was considered in the spherical P-spin glass. It is possible to store information in the vacuum configuration of ferromagnetic phase. Maximal allowed level of noise was calculated in ferromagnetic phase. Derrida's model [1,2] has been applied for optimal coding [3-6]. One chooses couplings

We consider an interacting particle system on a graph which, from a macroscopic point of view, looks like Z d and, at a microscopic level, is a complete graph of degree N (called a patch). There are two birth rates: an inter-patch one λ and an intra-patch one φ. Once a site is occupied, there is no breeding from outside the patch and the probability c(i) of success of an intra-patch breeding decreases with the size i of the population in the site. We prove the existence of a critical value λcr(φ, c, N) and a critical value φcr(λ, c, N). We consider a sequence of processes generated by the families of control functions {cn} n∈N and degrees {Nn} n∈N ; we prove, under mild assumptions, the existence of a critical value ncr(λ, φ, c). Roughly speaking we show that, in the limit, these processes behave as the branching random walk on Z d with inter-neighbor birth rate λ and on-site birth rate φ. Some examples of models that can be seen as particular cases are given.

In [18], Farrell and Whitehead investigate circulant graphs that are uniquely characterized by their matching and chromatic polynomials (i.e., graphs that are "matching unique" and "chromatic unique"). They develop a partial classification theorem, by finding all matching unique and chromatic unique circulants on n vertices, for each n ≤ 8. In this paper, we explore circulant graphs that are uniquely characterized by their independence polynomials. We obtain a full classification theorem by proving that a circulant is independence unique iff it is the disjoint union of isomorphic complete graphs.

Given a graph and a root, the Maximum Bounded Rooted-Tree Packing (MBRTP) problem aims at finding K rooted-trees that span the largest subset of vertices, when each vertex has a limited outdegree. This problem is motivated by peerto-peer streaming overlays in under-provisioned systems. We prove that the MBRTP problem is NP-complete. We present two polynomial-time algorithms that computes an optimal solution on complete graphs and trees respectively.

The Caccetta-Häggkvist conjecture (denoted below CHC) states that the directed girth (the smallest length of a directed cycle) dgirth(D) of a directed graph D on n vertices is at most ⌈ n δ + (D) ⌉, where δ + (D) is the minimum out-degree of D. We consider a version involving all out-degrees, not merely the minimum one, and prove that if D does not contain a sink, then dgirth(D) ≤ 2 v∈V (D) 1 deg + (v)+1. In the spirit of a generalization of the CHC to rainbow cycles in [1], this suggests the conjecture that given nonempty sets F 1 ,. .. , Fn of edges of Kn, there exists a rainbow cycle of length at most 2 1≤i≤n 1 |F i |+1. We prove a bit stronger result when 1 ≤ |F i | ≤ 2, thereby strengthening a result of DeVos et. al [6]. We prove a logarithmic bound on the rainbow girth in the case that the sets F i are triangles.

The Caccetta-Häggkvist conjecture (denoted below CHC) states that the directed girth (the smallest length of a directed cycle) dgirth(D) of a directed graph D on n vertices is at most ⌈ n δ + (D) ⌉, where δ + (D) is the minimum out-degree of D. We consider a version involving all out-degrees, not merely the minimum one, and prove that if D does not contain a sink, then dgirth(D) ≤ 2 v∈V (D) 1 deg + (v)+1. In the spirit of a generalization of the CHC to rainbow cycles in [1], this suggests the conjecture that given nonempty sets F 1 ,. .. , Fn of edges of Kn, there exists a rainbow cycle of length at most 2 1≤i≤n 1 |F i |+1. We prove a bit stronger result when 1 ≤ |F i | ≤ 2, thereby strengthening a result of DeVos et. al [6]. We prove a logarithmic bound on the rainbow girth in the case that the sets F i are triangles.

A two-valued function f defined on the vertices of a graph G = (V, E), f : V → {-1, 1}, is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. That is, for every v ∈ V, f (N (v)) ≥ 1, where N (v) consists of every vertex adjacent to v. The weight of a total signed dominating function is f (V ) = f (v), over all vertices v ∈ V . The total signed domination number of a graph G, denoted γ s t (G), equals the minimum weight of a total signed dominating function of G. If, instead of the range {-1, 1}, we allow the range {-1, 0, 1}, then we get the concept of a total minus dominating function. Its associated parameter, called the total minus domination number of a graph G, is denoted γ - t (G). In this paper, we show that the decision problem corresponding to the computation of the total minus domination number of a graph is NP-complete, even when restricted to bipartite graphs or chordal graphs. For a fixed k, we show that the decision problem corresponding to determining whether a graph has a total minus dominating function of weight at most k may be NPcomplete, even when restricted to bipartite or chordal graphs. Linear time algorithms for computing γ - t (T ) and γ s t (T ) for an arbitrary tree T are also presented.

Let K,, be the complete graph with vertex set {u,, u2,. . . , u,,) and let g = (gl, . . . , g,,) be a sequence of positive integers. Color each edge of this K, red or blue. In this paper necessary and sufficient conditions are given which guarantee the existence of a connected span&g subgraph F in K,, (as colored) with both red degree and blue degree in F at vertex Ui eqa$ to gi. When each & = 1 this answers a question of Erdiis proved in this special case in [I].

Let G be a subgraph of K(n), the complete graph on n vertices, such that (i) its edges cannot be represented by fewer than k vertices and (ii) every hamiltonian cycle of K(n) contains at least one edge of G and no proper subgraph of G has this property. P. Erdos posed the question of determining mine(G). In particular, is there an absolute constant c such that e(G) 2 c' k. n? The minimum is calculated for all n> 3 and k<n/2, and the second part of the question is answered in affirmative with c = 4 being the best possible constant.

For integers r ≥ 2 and s ≥ 1, let K r×s denote the complete multipartite graph with r partite sets of order s. Let G be a 2-regular graph of odd order n. If G contains exactly one odd cycle, it is known that there exists a G-decomposition of K 2kn+1 , of K (2k+1)×n , and of K k ×2n for all positive integers k and k ≥ 3. If G consists of three vertex-disjoint odd cycles, then the only known general result is a G-decomposition of K 2n+1. We use a novel extension of the Bose construction for triple systems to show that in the three odd cycles case, there exists a G-decomposition of K (2k+1)×n for every positive integer k. We also show that there exists a G-decomposition of K k×2n as well as of K 2kn+1 for every integer k ≥ 3.

We consider the decomposition of the complete bipartite graph K'm,n into isomorphic copies of a d-cube. We present some general necessary conditions for such a decomposition and show that these conditions are sufficient for d 3 and d = 4. We also explore the d-cube decomposition of the complete graph Kn. Necessary and sufficient conditions for the existence of such a decomposition are known for d even and for d odd and n odd. We present a general strategy for constructing these decompositions for all values of d. We use this method to show that the necessary conditions are sufficient for d = 3.

Let e 1 ; e 2 ; y; e n be a sequence of nonnegative integers such that the first non-zero term is not one. Let P n i¼1 e i ¼ ðq À 1Þ=2; where q ¼ p n and p is an odd prime. We prove that the complete graph on q vertices can be decomposed into e 1 C p n -factors, e 2 C p nÀ1 -factors, y; and e n C p -factors.

We prove a theorem about the decomposition of certain n-regular Cayley graphs into any tree with n edges. This result implies that the product of any r cycles of even length and the cube Qs decomposes into copies of any tree with 2r + sedges.

Navier-Stokes equations arise in the study of incompressible fluid mechanics, star movement inside a galaxy, dynamics of airplane wings, etc. In the case of Newtonian incompressible fluids, we propose an adaptation of such equations to finite connected weighted graphs such that it produces an ordinary differential equation with solutions contained in a linear subspace, this subspace corresponding to the Newtonian conservation law. We discuss the particular case when the graph is the complete graph K m , with constant weight, and provide a necessary and sufficient condition for it to have solutions.

A graph Γ is k-connected-homogeneous (k-CH) if k is a positive integer and any isomorphism between connected induced subgraphs of order at most k extends to an automorphism of Γ, and connected-homogeneous (CH) if this property holds for all k. We classify the locally finite 4-CH graphs that are either locally connected, or locally disconnected with girth 3 and c_2≠ 1. We then prove that, with the possible exception of locally disconnected graphs with girth 3 and c_2=1, every finite 7-CH graph is CH.

In this paper, we examine the structure of vertex-and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parameters of the graphs in this family that reduce to complete graphs. Definition 1.1. We refer to a vertex-and edge-transitive strongly regular graph as a ve-srg. Requiring strongly regular graphs to be vertex-and edge-transitive may seem a very stringent condition. However, several important and well-known families of graphs are contained within the class of ve-srgs,

In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g≥2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2–5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s≡1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3–5.

Usually dimension should be an integer valued parameter, we introduce a refined version of dimension for graphs which can assume a value [ t − 1 t ] which is thought to be between t − 1 and t. We have the following two results: • A graph is outerplanar if and only if its dimension is at most [ 2 3 ]. This characterization of outerplanar graphs is closely related to the celebrated result of W. Schnyder [16] who proved that a graph is planar if and only if its dimension is at most 3. • The largest n for which the dimension of the complete graph Kn is at most [ t − 1 t ] is the number of antichains in the lattice of all subsets of a set of size t − 2. Accordingly, the refined dimension problem for complete graphs is equivalent to the classical combinatorial problem known as Dedekind's problem. This result extends work of Hoşten and Morris [14]. The main results are enriched by background material which links to a line of reserch in extremal graph theory which was stimulated by a problem posed by G. Agnarsson: Find the maximum number of edges in a graph on n nodes with dimension at most t.

Given a digraph G and a su ciently long directed path P, a folklore result says that G is homomorphic to P if and only if all cycles in G are balanced (the same number of forward and backward arcs). The purpose of this paper is to study homomorphisms of digraphs G that contain unbalanced cycles. In this case, we may be able to ÿnd a homomorphism of G to a power of P. Our main result states that the minimum power of P to which G admits a homomorphism equals the maximum imbalance (ratio of forward and backward arcs) of any cycle in G. The proof also yields a polynomial time algorithm to ÿnd this minimum power of P. We identify a larger class of digraphs H for which this minimum power problem can be solved in polynomial time, it includes all oriented paths H. By relating our powers of paths to complete graphs and so-called circular graphs, we are able to deduce a classical result of Minty regarding the chromatic number, as well as its more recent extension, by Goddyn, Tarsi, and Zhang, to the circular chromatic number.