Universal targets for homomorphisms of edge-colored graphs

A $k$-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots,k$. A function from the vertex set of one $k$-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class $\mathcal{F}$ of graphs, a $k$-edge-colored graph $\mathbb{H}$ (not necessarily with the underlying graph in $\mathcal{F}$) is $k$-universal for $\mathcal{F}$ when any $k$-edge-colored graph with the underlying graph in $\mathcal{F}$ admits a homomorphism to $\mathbb{H}$. We characterize graph classes that admit $k$-universal graphs. For such classes, we establish asymptotically almost tight bounds on the size of the smallest universal graph. For a nonempty graph $G$, the density of $G$ is the maximum ratio of the number of edges to the number of vertices ranging over all nonempty subgraphs of $G$. For a nonempty class $\mathcal{F}$ of graphs, $D(\mathcal{F})$ denotes the density of $\mathcal{F}$, that is the supremum of densities of graphs in $\mathcal{F}$. The main results are the following. The class $\mathcal{F}$ admits $k$-universal graphs for $k\geq2$ if and only if there is an absolute constant that bounds the acyclic chromatic number of any graph in $\mathcal{F}$. For any such class, there exists a constant $c$, such that for any $k \geq 2$, the size of the smallest $k$-universal graph is between $k^{D(\mathcal{F})}$ and $ck^{\lceil D(\mathcal{F})\rceil}$. A connection between the acyclic coloring and the existence of universal graphs was first observed by Alon and Marshall (Journal of Algebraic Combinatorics, 8(1):5-13, 1998). One of their results is that for planar graphs, the size of the smallest $k$-universal graph is between $k^3+3$ and $5k^4$. Our results yield that there exists a constant $c$ such that for all $k$, this size is bounded from above by $ck^3$

A Dichotomy Theorem for Homomorphism Polynomials

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over the rational field of cut eliminator, a polynomial defined by B\"urgisser which is known to be neither VP nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is the class of polynomials computable by arithmetic circuit of polynomial size)

Persistent Homology Over Directed Acyclic Graphs

We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source and a single sink in $O(n^4)$ arithmetic operations, where $n$ is the number of vertices in the graph. We then demonstrate as an application of these tools a method to overlay two distinct filtrations of the same underlying space, which allows us to detect the most significant barcodes using considerably fewer points than standard persistence.Comment: Revised versio

Faster Subgraph Counting in Sparse Graphs

A fundamental graph problem asks to compute the number of induced copies of a k-node pattern graph H in an n-node graph G. The fastest algorithm to date is still the 35-years-old algorithm by Nesetril and Poljak [Nesetril and Poljak, 1985], with running time f(k) * O(n^{omega floor[k/3] + 2}) where omega <=2.373 is the matrix multiplication exponent. In this work we show that, if one takes into account the degeneracy d of G, then the picture becomes substantially richer and leads to faster algorithms when G is sufficiently sparse. More precisely, after introducing a novel notion of graph width, the DAG-treewidth, we prove what follows. If H has DAG-treewidth tau(H) and G has degeneracy d, then the induced copies of H in G can be counted in time f(d,k) * O~(n^{tau(H)}); and, under the Exponential Time Hypothesis, no algorithm can solve the problem in time f(d,k) * n^{o(tau(H)/ln tau(H))} for all H. This result characterises the complexity of counting subgraphs in a d-degenerate graph. Developing bounds on tau(H), then, we obtain natural generalisations of classic results and faster algorithms for sparse graphs. For example, when d=O(poly log(n)) we can count the induced copies of any H in time f(k) * O~(n^{floor[k/4] + 2}), beating the Nesetril-Poljak algorithm by essentially a cubic factor in n

D-colorable digraphs with large girth

In 1959 Paul Erdos (Graph theory and probability, Canad. J. Math. 11 (1959), 34-38) famously proved, nonconstructively, that there exist graphs that have both arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants (D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll and B. Mohar, The circular chromatic number of a digraph, J. Graph Theory 46 (2004), 227-240; B. Bollobas and N. Sauer, Uniquely colourable graphs with large girth, Canad. J. Math. 28 (1976), 1340-1344; J. Nesetril and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B 90 (2004), 161-172; X. Zhu, Uniquely H-colorable graphs with large girth, J. Graph Theory 23 (1996), 33-41) that have extended and generalized the result while strengthening the techniques used to achieve it. We follow the lead of Xuding Zhu (op. cit.) who proved that, for a suitable graph H, there exist graphs of arbitrarily large girth that are uniquely H-colorable. We establish an analogue of Zhu\u27s results in a digraph setting. Let C and D be digraphs. A mapping f:V(D)&rarr V(C) is a C-coloring if for every arc uv of D, either f(u)f(v) is an arc of C or f(u)=f(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colorable if it admits a C-coloring and that D is uniquely C-colorable if it is surjectively C-colorable and any two C-colorings of D differ by an automorphism of C. We prove that if D is a digraph that is not C-colorable, then there exist graphs of arbitrarily large girth that are D-colorable but not C-colorable. Moreover, for every digraph D that is uniquely D-colorable, there exists a uniquely D-colorable digraph of arbitrarily large girth