A mapping of $k$-bit strings into $n$-bit strings is called an $(α,β)$-map if $k$-bit strings which are more than $αk$ apart are mapped to $n$-bit strings that are more than $βn$ apart. This is a relaxation of the classical problem of constructing error-correcting codes, which corresponds to $α=0$. Existence of an $(α,β)$-map is equivalent to existence of a graph homomorphism $\bar H(k,αk)\to \bar H(n,βn)$, where $H(n,d)$ is a Hamming graph with vertex set $\{0,1\}^n$ and edges connecting vertices differing in $d$ or fewer entries.
This paper proves impossibility results on achievable parameters $(α,β)$ in the regime of $n,k\to\infty$ with a fixed ratio ${n\over k}= ρ$. This is done by developing a general criterion for existence of graph-homomorphism based on the semi-definite relaxation of the independence number of a graph (known as the Schrijver's $θ$-function). The criterion is then evaluated using some known and some new results from coding theory concerning the $θ$-function of Hamming graphs. As an example, it is shown that if $β>1/2$ and $n\over k$ -- integer, the ${n\over k}$-fold repetition map achieving $α=β$ is asymptotically optimal.
Finally, constraints on configurations of points and hyperplanes in projective spaces over $\mathbb{F}_2$ are derived.
Write ${\cal F}$ for the set of homomorphisms from $\{0,1\}^d$ to ${\bf Z}$ which send $\underline{0}$ to 0 (think of members of ${\cal F}$ as labellings of $\{0,1\}^d$ in which adjacent strings get labels differing by exactly 1), and ${\cal F}_i$ for those which take on exactly $i$ values. We give asymptotic formulae for $|{\cal F}|$ and $|{\cal F}_i|$. In particular, we show that the probability that a uniformly chosen member ${\bf f}$ of ${\cal F}$ takes more than five values tends to 0 as $d \rightarrow \infty$. This settles a conjecture of J. Kahn. Previously, Kahn had shown that there is…
We show that for each d>0 the d-dimensional Hamming graph H(d,q) has an orientably regular surface embedding if and only if q is a prime power p^e. If q>2 there are up to isomorphism ϕ(q-1)/e such maps, all constructed as Cayley maps for a d-dimensional vector space over the field of order q. We show that for each such pair d, q the corresponding Belyi pairs are conjugate under the action of the absolute Galois group, and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for q>3.
A function $f: \mathbb{Z} \to \mathbb{Q}^n$ is a $c$-quasihomomorphism if the Hamming distance between $f(x+y)$ and $f(x)+f(y)$ is at most $c$ for all $x,y \in \mathbb{Z}$. We show that any $c$-quasihomomorphism has distance at most some constant $C(c)$ to an actual group homomorphism; here $C(c)$ depends only on $c$ and not on $n$ or $f$. This gives a positive answer to a special case of a question posed by Kazhdan and Ziegler.
This paper presents a method to find new De Bruijn cycles based on ones of lesser order. This is done by mapping a De Bruijn cycle to several vertex disjoint cycles in a De Bruijn digraph of higher order and connecting these cycles into one full cycle. We characterize homomorphisms between De Bruijn digraphs of different orders that allow this construction. These maps generalize the well-known D-morphism of Lempel between De Bruijn digraphs of consecutive orders. Also, an efficient recursive algorithm that yields an exponential number of nonbinary De Bruijn cycles is implemented.
Graph embeddings deal with injective maps from a given simple, undirected graph $G=(V,E)$ into a metric space, such as $\mathbb{R}^n$ with the Euclidean metric. This concept is widely studied in computer science, see \cite{ge1}, but also offers attractive research in pure graph theory \cite{ge2}. In this note we show that any graph can be embedded into a particularly simple metric space: $\{0,1\}^n$ with the Hamming distance, for large enough $n$.
We extend the notion of 'homomorphism-homogeneity' to a wider class of kinds of maps than previously studied, and we investigate the relations between the resulting notions of homomorphism-homogeneity, giving several examples. We also give further details on related work reported in [Deborah Lockett and John K Truss, Generic endomorphisms of homogeneous structures, in 'Groups and model theory', Contemporary Mathematics 576, ed Strungmann, Droste, Fuchs, Tent, American Mathematical Society, 2012, 217-237] about the endomorphisms of chains and generic endomorphisms of tree.
For graphs $G$ and $H$, a homomorphism from $G$ to $H$ is a function $φ\colon V(G) \to V(H)$, which maps vertices adjacent in $G$ to adjacent vertices of $H$. A homomorphism is locally injective if no two vertices with a common neighbor are mapped to a single vertex in $H$. Many cases of graph homomorphism and locally injective graph homomorphism are NP-complete, so there is little hope to design polynomial-time algorithms for them. In this paper we present an algorithm for graph homomorphism and locally injective homomorphism working in time $\mathcal{O}^*((b + 2)^{|V(G)|})$, where $b$ is the…
We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tension-continuous mappings (shortly TT mappings). Existence of a TT mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied extensively, see [Hell-Nesetril]). In this paper we study the relationship of the homomorphism and TT orders. We stress the similarities and the differences in both deterministic and random setting. Particularly, we prove that TT order is dense and universal and we solve a problem of M. DeVos …
RIKEN Center for Advanced Intelligence Project（日本理化学研究所先进智能项目）；Tokyo Institute of Technology（东京工业大学）
In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some family of graphs (e.g. paths or non-isomorphic trees). We show that graph homomorphism numbers provide a natural invariant (isomorphism invariant and $\mathcal{F}$-invariant) embedding maps which can be used for graph classification. Viewing the expressive power of a graph classifier by the $\mathcal{F}$-indistinguishable concept, we prove the universality pro…
We introduce (weak) oddomorphisms of graphs which are homomorphisms with additional constraints based on parity. These maps turn out to have interesting properties (e.g., they preserve planarity), particularly in relation to homomorphism indistinguishability. Graphs $G$ and $H$ are *homomorphism indistinguishable* over a family $\mathcal{F}$ if $\hom(F,G) = \hom(F,H)$ for all $F \in \mathcal{F}$, where $\hom(F,G)$ is the number of homomorphisms from $F$ to $G$. A classical result of Lovász says that isomorphism is equivalent to homomorphism indistinguishability over the class of all graphs. In…
A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is `sub-logarithmic', then the range of such a homomorphism is super-constant. Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of gr…
The complexity of graph homomorphisms has been a subject of intense study [11, 12, 4, 42, 21, 17, 6, 20]. The partition function $Z_{\mathbf A}(\cdot)$ of graph homomorphism is defined by a symmetric matrix $\mathbf A$ over $\mathbb C$. We prove that the complexity dichotomy of [6] extends to bounded degree graphs. More precisely, we prove that either $G \mapsto Z_{\mathbf A}(G)$ is computable in polynomial-time for every $G$, or for some $Δ> 0$ it is #P-hard over (simple) graphs $G$ with maximum degree $Δ(G) \le Δ$. The tractability criterion on $\mathbf A$ for this dichotomy is explicit, and…
Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural …
Given finite posets $P$ and $Q$, we consider a specific ideal $L(P,Q)$, whose minimal monomial generators correspond to order-preserving maps $ϕ:P\rightarrow Q$. We study algebraic invariants of those ideals. In particular, sharp lower and upper bounds for the Castelnuovo-Mumford regularity and the projective dimension are provided. Precise formulas are obtained for a large subclass of these ideals. Moreover, we provide complete characterizations for several algebraic properties of $L(P,Q)$, including being Buchsbaum, Cohen-Macaulay, Gorenstein and having a linear resolution. We also give a pa…
Over 50 years ago, Lovász proved that two graphs are isomorphic if and only if they admit the same number of homomorphisms from any graph [Acta Math. Hungar. 18 (1967), pp. 321--328]. In this work we prove that two graphs are quantum isomorphic (in the commuting operator framework) if and only if they admit the same number of homomorphisms from any planar graph. As there exist pairs of non-isomorphic graphs that are quantum isomorphic, this implies that homomorphism counts from planar graphs do not determine a graph up to isomorphism. Another immediate consequence is that determining whether t…
