Given an undirected graph $G$ and integers $c$ and $k$, the Maximum Edge-Colorable Subgraph problem asks whether we can delete at most $k$ edges in $G$ to obtain a graph that has a proper edge coloring with at most $c$ colors. We show that Maximum Edge-Colorable Subgraph admits, for every fixed $c$, a linear-size problem kernel when parameterized by the edge deletion distance of $G$ to a graph with maximum degree $c-1$. This parameterization measures the distance to instances that, due to Vizing's famous theorem, are trivial yes-instances. For $c\le 4$, we also provide a linear-size kernel for the same parameterization for Multi Strong Triadic Closure, a related edge coloring problem with applications in social network analysis. We provide further results for Maximum Edge-Colorable Subgraph parameterized by the vertex deletion distance to graphs where every component has order at most $c$ and for the list-colored versions of both problems.
Inductive $k$-independent graphs generalize chordal graphs and have recently been advocated in the context of interference-avoiding wireless communication scheduling. The NP-hard problem of finding maximum-weight induced $c$-colorable subgraphs, which is a generalization of finding maximum independent sets, naturally occurs when selecting $c$ sets of pairwise non-conflicting jobs (modeled as graph vertices). We investigate the parameterized complexity of this problem on inductive $k$-independent graphs. We show that the Independent Set problem is W[1]-hard even on 2-simplicial 3-minoes---a sub…
Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the Strong F-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In Strong F-closure, we want to select a maximum number of edges of the input graph G, and mark them as strong edges, in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F, then the corresponding induced subgrap…
Let $G=(V,E)$ be a multigraph of maximum degree $Δ$. The edges of $G$ can be colored with at most $\frac{3}{2}Δ$ colors by Shannon's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $Δ$ colors. Shannon's Theorem gives a bound of $\fracΔ{\lfloor\frac{3}{2}Δ\rfloor}|E|$. However, for $Δ=3$, Kamiński and Kowalik [SWAT'10] showed that there is a 3-edge-colorable subgraph of size at least $\frac{7}{9}|E|$, unless $G$ has a connected component isomorphic to $K_3+e$ (a $K_3$ with an arbitrary edge doubled). Here we extend this line of research by showing that $G…
We study the \textsc{Max Partial $H$-Coloring} problem: given a graph $G$, find the largest induced subgraph of $G$ that admits a homomorphism into $H$, where $H$ is a fixed pattern graph without loops. Note that when $H$ is a complete graph on $k$ vertices, the problem reduces to finding the largest induced $k$-colorable subgraph, which for $k=2$ is equivalent (by complementation) to \textsc{Odd Cycle Transversal}. We prove that for every fixed pattern graph $H$ without loops, \textsc{Max Partial $H$-Coloring} can be solved: $\bullet$ in $\{P_5,F\}$-free graphs in polynomial time, whenever $F…
It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs, and hence 3-edge-colorable. We provide an efficient recognition algorithm that given an $n$-vertex planar graph, augments this graph in $O(n^2)$ steps to a planar cubic bridgeless supergraph, or decides that no such augmentation is possible. The main tools involve the Generalized Antifactor-problem for the fixed embedding case, and SPQR-trees for the variab…
A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. Th…
Let $G=(V,E)$ be a simple graph of maximum degree $Δ$. The edges of $G$ can be colored with at most $Δ+1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $Δ$ colors. Vizing's Theorem gives a bound of $\fracΔ{Δ+1}|E|$. This is known to be tight for cliques $K_{Δ+1}$ when $Δ$ is even. However, for $Δ=3$ it was improved to $26/31|E|$ by Albertson and Haas [Parsimonious edge colorings, Disc. Math. 148, 1996] and later to $6/7|E|$ by Rizzi [Approximating the maximum 3-edge-colorable subgraph problem, Disc. Math. 309, 2009]. It is tight for …
Improved SDP Coloring of 3-Colorable Graphs from Recursive Gaussian Certificates
Georgia Institute of Technology, School of Computer Science（佐治亚理工学院计算机学院）；Princeton University, Operations Research and Financial Engineering（普林斯顿大学运筹与金融工程学系）
Institute for Theoretical Physics, University of Würzburg（维尔茨堡大学理论物理研究所）
The detection of triadic subgraph motifs is a common methodology in complex-networks research. The procedure usually applied in order to detect motifs evaluates whether a certain subgraph pattern is overrepresented in a network as a whole. However, motifs do not necessarily appear frequently in every region of a graph. For this reason, we recently introduced the framework of Node-Specific Pattern Mining (NoSPaM). This work is a manual for an implementation of NoSPaM which can be downloaded from www.mwinkler.eu.
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no $(k-ε)^{\operatorname{pw}(G)}\operatorname{poly}(n)$ time algorithm for deciding if an $n$-vertex graph $G$ with pathwidth $\operatorname{pw}(G)$ admits a proper vertex coloring with $k$ colors unless the Strong Exponential Time Hypothesis (SETH) is false. We show here that nevertheless, when $k>\lfloor Δ/2 \rfloor + 1$, where $Δ$ is the maximum degree in the graph $G$, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph $G$ along with a path decomposition of $G…
Finding large cliques or cliques missing a few edges is a fundamental algorithmic task in the study of real-world graphs, with applications in community detection, pattern recognition, and clustering. A number of effective backtracking-based heuristics for these problems have emerged from recent empirical work in social network analysis. Given the NP-hardness of variants of clique counting, these results raise a challenge for beyond worst-case analysis of these problems. Inspired by the triadic closure of real-world graphs, Fox et al. (SICOMP 2020) introduced the notion of $c$-closed graphs an…
Poljak and Turzík (Discrete Math. 1986) introduced the notion of λ-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0<λ<1 and λ-extendible property Π, any connected graph G on n vertices and m edges contains a subgraph H \in Π with at least λm+ (1-λ)/2 (n-1) edges. The property of being bipartite is 1/2-extendible, and thus this bound generalizes the Edwards-Erdős bound for Max-Cut. We define a variant, namely strong λ-extendibility, to which the bound applies. For a strongly λ-extendible graph property Π, we define the parameteri…
Maximum Partial List H-Coloring on P_5-free graphs in polynomial time
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Warsaw University of Technology（华沙理工大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；Institute of Mathematical Sciences, Chennai（金奈数学科学研究所）；University of Bergen（卑尔根大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究院离散数学组）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）
A widely studied problem in communication networks is that of finding the maximum number of communication requests that can be scheduled concurrently, subject to node and/or link capacity constraints. In this paper, we consider the problem of finding the largest number of multicast communication requests that can be serviced simultaneously by a network of tree topology, subject to heterogeneous capacity constraints. This problem generalizes the following two problems studied in the literature: a) the problem of finding a largest induced $k$-colorable subgraph of a chordal graph, b) the maximum…
A fundamental graph problem is to recognize whether the vertex set of a graph $G$ can be bipartitioned into sets $A$ and $B$ such that $G[A]$ and $G[B]$ satisfy properties $Π_A$ and $Π_B$, respectively. This so-called $(Π_A,Π_B)$-Recognition problem generalizes amongst others the recognition of $3$-colorable, bipartite, split, and monopolar graphs. In this paper, we study whether certain fixed-parameter tractable $(Π_A,Π_B)$-Recognition problems admit polynomial kernels. In our study, we focus on the first level above triviality, where $Π_A$ is the set of $P_3$-free graphs (disjoint unions of …
