Subgraph Containment Research Articles

Rainbow spanning structures in graph and hypergraph systems

Abstract We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection $\mathit {\mathbf {G}}=\{G_1, G_2,\ldots , G_{m}\}$ of not necessarily distinct k-graphs on the same vertex set $[n]$ , a (sub)graph H on $[n]$ is rainbow if there exists an injection $\varphi : E(H)\rightarrow [m]$ , such that $e\in E(G_{\varphi (e)})$ for each $e\in E(H)$ . Note that if $|E(H)|=m$ , then $\varphi $ is a bijection, and thus H contains exactly one edge from each $G_i$ . Our main results focus on rainbow clique-factors in (hyper)graph systems with minimum d-degree conditions. Specifically, we establish the following: (1) A rainbow analogue of an asymptotical version of the Hajnal–Szemerédi theorem, namely, if $t\mid n$ and $\delta (G_i)\geq (1-\frac {1}{t}+\varepsilon )n$ for each $i\in [\frac {n}{t}\binom {t}{2}]$ , then $\mathit {\mathbf {G}}$ contains a rainbow $K_t$ -factor; (2) Essentially, a minimum d-degree condition forcing a perfect matching in a k-graph also forces rainbow perfect matchings in k-graph systems for $d\in [k-1]$ . The degree assumptions in both results are asymptotically best possible (although the minimum d-degree condition forcing a perfect matching in a k-graph is in general unknown). For (1), we also discuss two directed versions and a multipartite version. Finally, to establish these results, we in fact provide a general framework to attack this type of problem, which reduces it to subproblems with finitely many colors.

HyperISO: Efficiently Searching Subgraph Containment in Hypergraphs

HyperISO: Efficiently Searching Subgraph Containment in Hypergraphs

Deciding whether a grid is a topological subgraph of a planar graph is NP-complete

The Topological Subgraph Containment (TSC) Problem is to decide, for two given graphs G and H, whether H is a topological subgraph of G. It is known that the TSC Problem is NP-complete when H is part of the input, that it can be solved in polynomial time when H is fixed, and that it is fixed-parameter tractable by the order of H.Motivated by the great significance of grids in graph theory and algorithms due to the Grid-Minor Theorem by Robertson and Seymour, we investigate the computational complexity of the Grid TSC Problem in planar graphs. More precisely, we study the following decision problem: given a positive integer k and a planar graph G, is the k×k grid a topological subgraph of G? We prove that this problem is NP-complete, even when restricted to planar graphs of maximum degree six, via a novel reduction from the Planar Monotone 3-SAT Problem.

On the probability of nonexistence in binomial subsets

Given a hypergraph $\Gamma =(\Omega ,\mathcal{X})$ and a sequence $\mathbf{p}=(p_{\omega })_{\omega \in \Omega }$ of values in $(0,1)$, let $\Omega_{\mathbf{p}}$ be the random subset of $\Omega $ obtained by keeping every vertex $\omega $ independently with probability $p_{\omega }$. We investigate the general question of deriving fine (asymptotic) estimates for the probability that $\Omega_{\mathbf{p}}$ is an independent set in $\Gamma $, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of upper and lower bounds on this probability, each of which can be evaluated explicitly in terms of the joint cumulants of small sets of edge indicator random variables. Under certain natural conditions, these upper and lower bounds coincide asymptotically, thus giving the precise asymptotics of the probability in question. We demonstrate the applicability of our results with two concrete examples: subgraph containment in random (hyper)graphs and arithmetic progressions in random subsets of the integers.

Deciding Whether a Grid is a Topological Subgraph of a Planar Graph is NP-Complete

The Topological Subgraph Containment (TSC) Problem is to decide, for two given graphs G and H, whether H is a topological subgraph of G. It is known that the TSC Problem is NP-complete when H is part of the input, that it can be solved in polynomial time when H is fixed, and that it is fixed-parameter tractable by the order of H.Motivated by the great significance of grids in graph theory and algorithms due to the Grid-Minor Theorem by Robertson and Seymour, we investigate the computational complexity of the Grid TSC Problem in planar graphs. More precisely, we study the following decision problem: given a positive integer k and a planar graph G, is the k × k grid a topological subgraph of G? We prove that this problem is NP-complete, even when restricted to planar graphs of maximum degree six, via a novel reduction from the Planar Monotone 3-SAT Problem.

Every Graph $G$ is Hall $\Delta(G)$-Extendible

In the context of list coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list coloring. The graph $G$ with list assignment $L$, abbreviated $(G,L)$, satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $k$-extendible for some $k \geq \chi(G)$ if every $k$-precoloring of $G$ whose corresponding list assignment is Hall can be extended to a proper $k$-coloring of $G$. In 2011, Bobga et al. posed the question: If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-extendible? This paper establishes an affirmative answer to this question: every graph $G$ is Hall $\Delta(G)$-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of $k$ for which they are Hall $k$-extendible is presented. We include a focus on graphs which are Hall $k$-extendible for all $k \geq \chi(G)$, since these are graphs for which satisfying the obviously necessary Hall's condition is also sufficient for a precoloring to be extendible.

QUANTUM QUERY COMPLEXITY OF CONSTANT-SIZED SUBGRAPH CONTAINMENT

We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether a n-vertex graph contains a triangle, clique or star of some size. For a general subgraph H with k vertices, we show that H containment can be solved with quantum query complexity [Formula: see text], with g(H) a strictly positive function of H. This is better than Õ(n2-2/k) by Magniez et al. This result is obtained in the learning graph model of Belovs.

Vertex-colouring edge-weightings with two edge weights

Graphs and Algorithms An edge-weighting vertex colouring of a graph is an edge-weight assignment such that the accumulated weights at the vertices yields a proper vertex colouring. If such an assignment from a set S exists, we say the graph is S-weight colourable. It is conjectured that every graph with no isolated edge is \1, 2, 3\-weight colourable. We explore the problem of classifying those graphs which are \1, 2\ -weight colourable. We establish that a number of classes of graphs are S -weight colourable for much more general sets S of size 2. In particular, we show that any graph having only cycles of length 0 mod 4 is S -weight colourable for most sets S of size 2. As a consequence, we classify the minimal graphs which are not \1, 2\-weight colourable with respect to subgraph containment. We also demonstrate techniques for constructing graphs which are not \1, 2\-weight colourable.

Excluding induced subgraphs: Critical graphs

AbstractDetermining the cardinality and describing the structure of H‐free graphs is well‐investigated for many graphs H. In the nineties, Prömel and Steger proved that for a graph H with chromatic number k + 1 almost all graphs not containing H as a subgraph are k‐colorable if and only if H contains a color‐critical edge. We strengthen the concept of H‐free to induced subgraph containment, proving that if H has coloring number k + 1 then almost all H‐free graphs can be covered by k graphs that are cliques or independent sets if and only if H is in some well‐defined sense critical. The family of critical graphs includes C4 and C2k+1 for all k ≥ 3. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011

Evasiveness of Subgraph Containment and Related Properties

We prove new results on evasiveness of monotone graph properties by extending the techniques of Kahn, Saks, and Sturtevant [ Combinatorica, 4 (1984), pp. 297--306]. For the property of containing a subgraph isomorphic to a fixed graph, and a fairly large class of related n-vertex graph properties, we show evasiveness for an arithmetic progression of values of n. This implies a $\frac12n^2 - O(n)$ lower bound on the decision tree complexity of these properties. We prove that properties that are preserved under taking graph minors are evasive for all sufficiently large n. This greatly generalizes a theorem due to Best, van Emde Boas, and Lenstra [A Sharpened Version of the Aanderaa--Rosenberg Conjecture, Report ZW 30/74, Mathematisch Centrum, Amsterdam, The Netherlands, 1974] which states that planarity is evasive. We prove a similar result for bipartite subgraph containment.