We make progress on a generalization of the road (colouring) problem. The road problem was posed by Adler-Goodwyn-Weiss and solved by Trahtman. The generalization was posed, and solved in certain special cases, by Ashley-Marcus-Tuncel. We resolve two new families of cases, of which one generalizes the road problem and follows Trahtman's solution, and the other generalizes a result of Ashley-Marcus-Tuncel with a proof quite different from theirs. Along the way, we prove a universal property for the fiber product of certain graph homomorphisms, which may be of independent interest. We provide polynomial-time algorithms for relevant constructions and decision problems.
We present a edge-coloured analogue of the duality theorem for transitive tournaments and directed paths. Given a edge-coloured path $P$ whose edges alternate blue and red, we construct a edge-coloured graph $D$ so that for any edge-coloured graph $G$ $$ P \to G \Leftrightarrow G \not\to D. $$ The duals are simple to construct, in particular $|V(D)|=|V(P)|-1$.
In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not\to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair. This relation between paths and tournaments implies that a graph is $k$-colourable if and only if it admits an orientation with no directed path on more than $k$-vertices. In this work, for every undirected cycle $C$ we find an orientation $C_D$ and an oriented path $P_C$, such that $(P_C,C_D)$ is a dual…
In this paper, we state and prove two Fraïssé-style results that cover existence and uniqueness properties for twelve of the eighteen different notions of homomorphism-homogeneity as introduced by Lockett and Truss, and provide forward directions and implications for the remaining six cases. Following these results, we completely determine the extent to which the countable homogeneous undirected graphs (as classified by Lachlan and Woodrow) are homomorphism-homogeneous; we also provide some insight into the directed graph case.
For a fixed graph H, the H-Recoloring problem asks whether for two given homomorphisms from a graph G to H, we can transform one into the other by changing the image of a single vertex of G in each step and maintaining a homomorphism from G to H throughout. We extend an algorithm of Wrochna for H-Recoloring where H is a square-free loopless undirected graph to the more general setting of directed graphs. We obtain a polynomial-time algorithm for H-Recoloring in this setting whenever H is a loopless digraph that does not contain a 4-cycle of algebraic girth zero and whenever H is a reflexive di…
The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road colouring theorem to graphs with non-constant out-degree; we give reasons to believe that both of these conjectures are true. Our main results focus on two classes of graphs, proving both conjectures for one class of graphs and one of the conjectures for an additional class of graphs. We also present computer simulations that give some empirical evidence fo…
In Chapter 1 we fully characterise pairs of finite graphs which form a gap in the full homomorphism order. This leads to a simple proof of the existence of generalised duality pairs. We also discuss how such results can be carried to relational structures with unary and binary relations. In Chapter 2 we show a very simple and versatile argument based on divisibility which immediately yields the universality of the homomorphism order of directed graphs and discuss three applications. In chapter 3, we show that every interval in the homomorphism order of finite undirected graphs is either univer…
For an integer k >= 1, the k-th interlacing adjoint of a digraph G is the digraph i_k(G) with vertex-set V(G)^k, and arcs ((u_1, ..., u_k), (v_1, ..., v_k)) such that (u_i,v_i) \in A(G) for i = 1, ..., k and (v_i, u_{i+1}) \in A(G) for i = 1, ..., k-1. For every k we derive upper and lower bounds for the chromatic number of i_k(G) in terms of that of G. In particular, we find tight bounds on the chromatic number of interlacing adjoints of transitive tournaments. We use this result in conjunction with categorial properties of adjoint functors to derive the following consequence. For every integ…
The editing of a combinatorial object is the alteration of some of its elements such that the resulting object satisfies a certain fixed property. The edit problem for graphs, when the edges are added or deleted, was first studied independently by the authors and Kézdy [J. Graph Theory (2008), 58(2), 123--138] and by Alon and Stav [Random Structures Algorithms (2008), 33(1), 87--104]. In this paper, a generalization of graph editing is considered for multicolorings of the complete graph as well as for directed graphs. Specifically, the number of edge-recolorings sufficient to be performed on a…
This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective objects, unlike the category of directed multigraphs (i.e. quivers). The category of incidence hypergraphs is introduced as a "graph-like" remedy for the set-system issues so that hypergraphs may be studied by their locally graphic behavior via homomorphisms that allow an edge of the domain to be mapped into a subset of an edge in the codomain. Moreover, it is…
In this paper the concept of circular $r$-flows in a mono-directed signed graph $(G, σ)$ is introduced. That is a pair $(D, f)$, where $D$ is an orientation on $G$ and $f: E(G)\to (-r,r)$ satisfies that $|f(e)|\in [1, r-1]$ for each positive edge $e$ and $|f(e)|\in [0, \frac{r}{2}-1]\cup [\frac{r}{2}+1, r)$ for each negative edge $e$, and the total in-flow equals the total out-flow at each vertex. The circular flow index of a signed graph $(G, σ)$ with no positive bridge, denoted $Φ_c(G,σ)$, is the minimum $r$ such that $(G, σ)$ admits a circular $r$-flow. This is the dual notion of circular c…
Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80's, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation …
An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_v\cap T_v\neq \emptyset$ for every vertex $v$. Compared to well-known undirected intersection graphs like interval graphs and permutation graphs, not many algorithmic applications on intersection digraphs have been developed. Motivated by the successful story on algorithmic applications of intersection graphs using a graph width parameter called mim-width…
Lattice-based Cryptography is considered to have the characteristics of classical computers and quantum attack resistance. We will design various graphic lattices and matrix lattices based on knowledge of graph theory and topological coding, since many problems of graph theory can be expressed or illustrated by (colored) star-graphic lattices. A new pair of the leaf-splitting operation and the leaf-coinciding operation will be introduced, and we combine graph colorings and graph labellings to design particular proper total colorings as tools to build up various graphic lattices, graph homomorp…
A transshipment problem (G, d, λ) is modeled by a directed graph G = (V, E) with weighted vertices d = (d_v | v \in V) and directed edges λ= (λ_e | e \in E) interpreted as follows: G is a communication or transportation network, e.g., a pipeline; each edge e \in E is a one-way communication line, road or pipe of capacity λ_e, while every vertex v \in V is a node of production d_v > 0, consumption d_v < 0, or transition d_v = 0. A non-negative flow x = (x_e \mid e \in E) is called weakly feasible if for each v \in V the algebraic sum of flows, over all directed edges incident to v, equals d_v; …
