Multicolor and directed edit distance

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 K\'ezdy [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 any edge-colored complete graph to satisfy a given hereditary property is investigated. The theory for computing the edit distance is extended using random structures and so-called types or colored homomorphisms of graphs.Comment: 25 page

Removing induced powers of cycles from a graph via fewest edits

What is the minimum proportion of edges which must be added to or removed from a graph of density $p$ to eliminate all induced cycles of length $h$? The maximum of this quantity over all graphs of density $p$ is measured by the edit distance function, $\text{ed}_{\text{Forb}(C_h)}(p)$, a function which provides a natural metric between graphs and hereditary properties. Martin determined $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [0,1]$ when $h \in \{3, \ldots, 9\}$ and determined $\text{ed}_{\text{Forb}(C_{10})}(p)$ for $p \in [1/7, 1]$. Peck determined $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [0,1]$ for odd cycles, and for $p \in [ 1/\lceil h/3 \rceil, 1]$ for even cycles. In this paper, we fully determine the edit distance function for $C_{10}$ and $C_{12}$. Furthermore, we improve on the result of Peck for even cycles, by determining $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [p_0, 1/\lceil h/3 \rceil ]$, where $p_0 \leq c/h^2$ for a constant $c$. More generally, if $C_h^t$ is the $t$-th power of the cycle $C_h$, we determine $\text{ed}_{\text{Forb}(C_h^t)}(p)$ for all $p \geq p_0$ in the case when $(t+1) \mid h$, thus improving on earlier work of Berikkyzy, Martin and Peck.Comment: 17 page

The edit distance function: Forbidding induced powers of cycles and other questions

The edit distance between two graphs on the same labeled vertex set is defined to be the size of the symmetric difference of the edge sets. The edit distance between a graph, $G$, and a graph property, $\mathcal{H}$, is the minimum edit distance between $G$ and a graph in $\mathcal{H}$. The edit distance function of a graph property $\mathcal{H}$ is a function of $p\in [0,1]$ that measures, in the limit, the maximum normalized edit distance between a graph of density $p$ and $\mathcal{H}$. In this thesis, we address the edit distance function for the property of having no induced copy of $C_h^t$, the t^{\mbox{th}} power of the cycle of length $h$. For $h\geq 2t(t+1)+1$ and $h$ not divisible by $t+1$, we determine the function for all values of $p$. For $h\geq 2t(t+1)+1$ and $h$ divisible by $t+1$, the function is obtained for all but small values of $p$. We also obtain some results for smaller values of $h$, present alternative proofs of some important previous results using simple optimization techniques and discuss possible extension of the theory to hypergraphs

On the edit distance from a cycle- and squared cycle-free graph

The edit distance from a hereditary property is the fraction of edges in a graph that must be added or deleted for a graph to become a member of that hereditary property. Let Forb(Ch) and Forb(C2h) denote the hereditary properties containing graphs with no induced cycle or squared cycle on h vertices, respectively. The edit distance from Forb(Ch) is found for odd values of h, and the maximum edit distance is found for all values of h. The edit distance is found for Forb(C2h) for h = 8; 9; 10, and the maximum value is known for h = 11; 12, with partial results for other values of h

Spanning and induced subgraphs in graphs and digraphs

In this thesis, we make progress on three problems in extremal combinatorics, particularly in relation to finding large spanning subgraphs, and removing induced subgraphs. First, we prove a generalisation of a result of Komlós, Sárközy and Szemerédi and show that for $n$ sufficiently large, any $n$-vertex digraph with minimum semidegree at least $n/2 + o(n)$ contains a copy of every $n$-vertex oriented tree with underlying maximum degree at most $O(n/\log n)$. For our second result, we prove that when $k$ is an even integer and $n$ is sufficiently large, if $G$ is a $k$-partite graph with vertex classes $V_1, \ldots, V_k$ each of size $n$ and $\delta(G[V_i,V_{i+1}]) \geq (1 + 1/k) n/2$, then $G$ contains a transversal $C_k$-factor, that is, a $C_k$-factor in which each copy of $C_k$ contains exactly one vertex from each vertex class. In the case when $k$ is odd, we reduce the problem to proving that when $G$ is close to a specific extremal structure, it contains a transversal $C_k$-factor. This resolves a conjecture of Fischer for even $k$. Our third result falls into the theory of edit distances. Let $C_h^t$ be the $t$-th power of a cycle of length $h$, that is, a cycle of length $h$ with additional edges between vertices at distance at most $t$ on the cycle. Let $\text{Forb}(C_h^t)$ be the class of graphs with no induced copy of $C_h^t$. For $p \in [0,1]$, what is the minimum proportion of edges which must be added to or removed from a graph of density $p$ to eliminate all induced copies of $C_h^t$? The maximum of this quantity over all graphs of density $p$ is measured by the edit distance function, $\text{ed}_{\text{Forb}(C_h^t)}(p)$, a function which provides a natural metric between graphs and hereditary properties. For our third result, we determine $\text{ed}_{\text{Forb}(C_h^t)}(p)$ for all $p \geq p_0$ in the case when $(t+1) \mid h$, where $p_0 = \Theta(1/h^2)$, thus improving on earlier work of Berikkyzy, Martin and Peck