Browsing Waterloo Research by Subject "induced subgraph"
Now showing items 1-12 of 12
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A Counterexample to a Conjecture About Triangle-Free Induced Subgraphs of Graphs with Large Chromatic Number
(Elsevier ScienceDirect, 2023-01)We prove that for every n, there is a graph G with χ(G) ≥ n and ω(G) ≤ 3 such that every induced subgraph H of G with ω(H) ≤ 2 satisfies χ(H) ≤ 4.This disproves a well-known conjecture. Our construction is a digraph with ... -
Four-coloring P6-free graphs. I. Extending an excellent precoloring.
(Society for Industrial and Applied Mathematics, 2024)This is the first paper in a series whose goal is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, ... -
Four-coloring P6-free graphs. II. Finding an excellent precoloring.
(Society for Industrial and Applied Mathematics, 2024)This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced ... -
Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree.
(Elsevier, 2024-01)This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum ... -
Induced subgraphs and tree decompositions VII. Basic obstructions in H-free graphs.
(Elsevier, 2024-01)We say a class C of graphs is clean if for every positive integer t there exists a positive integer w(t) such that every graph in C with treewidth more than w(t) contains an induced subgraph isomorphic to one of the ... -
List-3-Coloring Ordered Graphs with a Forbidden Induced Subgraph
(Society for Industrial and Applied Mathematics, 2024-03)Abstract. The List-3-Coloring Problem is to decide, given a graph G and a list L(v) ⊆ {1, 2, 3} of colors assigned to each vertex v of G, whether G admits a proper coloring ϕ with ϕ(v) ∈ L(v) for every vertex v of G, and ... -
Polynomial bounds for chromatic number II: Excluding a star-forest
(Wiley, 2022-10)The Gyárfás–Sumner conjecture says that for every forest H, there is a function fH such that if G is H-free then x(G) ≤ fH(w(G)) (where x,w are the chromatic number and the clique number of G). Louis Esperet conjectured ... -
Polynomial bounds for chromatic number VII. Disjoint holes.
(Wiley, 2023-05-14)A hole in a graph G is an induced cycle of length at least four, and a k-multihole in G is the union of k pairwise disjoint and nonneighbouring holes. It is well known that if G does not contain any holes then its chromatic ... -
Polynomial bounds for chromatic number VII. Disjoint holes.
(Wiley, 2023-11)A hole in a graph G is an induced cycle of length at least four, and a k-multihole in G is the union of k pairwise disjoint and nonneighbouring holes. It is well know that if G does not contain any holes then its chromatic ... -
Pure pairs. IV. Trees in bipartite graphs.
(Elsevier, 2023-07)In this paper we investigate the bipartite analogue of the strong Erd˝os-Hajnal property. We prove that for every forest H and every τ with 0 < τ ≤ 1, there exists ε > 0, such that if G has a bipartition (A,B) and does not ... -
Pure pairs. VII. Homogeneous submatrices in 0/1-matrices with a forbidden submatrix
(Elsevier, 2023-07)For integer n>0, let f(n) be the number of rows of the largest all-0 or all-1 square submatrix of M, minimized over all n x n 0/1-matrices M. Thus f(n)=O(log n). But let us fix a matrix H, and define fH(n) to be the same, ... -
Triangle-free graphs with no six-vertex induced path
(Elsevier, 2018-08)The graphs with no five-vertex induced path are still not understood. But in the triangle-free case, we can do this and one better; we give an explicit construction for all triangle-free graphs with no six-vertex induced ...