UWSpaceThe UWSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.https://uwspace.uwaterloo.ca:4432022-08-12T08:28:13Z2022-08-12T08:28:13ZApproximately Coloring Graphs Without Long Induced PathsChudnovsky, MariaSchaudt, OliverSpirkl, Sophiestein, mayaZhong, Mingxianhttp://hdl.handle.net/10012/185362022-08-12T02:31:30Z2017-01-01T00:00:00ZApproximately Coloring Graphs Without Long Induced Paths
Chudnovsky, Maria; Schaudt, Oliver; Spirkl, Sophie; stein, maya; Zhong, Mingxian
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on t vertices, for fixed t. We propose an algorithm that, given a 3-colorable graph without an induced path on t vertices, computes a coloring with max{5,2⌈t−12⌉−2} many colors. If the input graph is triangle-free, we only need max{4,⌈t−12⌉+1} many colors. The running time of our algorithm is O((3t−2+t2)m+n) if the input graph has n vertices and m edges.
This is a post-peer-review, pre-copyedit version of a conference paper published in Lecture Notes in Computer Science. The final authenticated version is available online at: https://doi.org/10.1007/978-3-319-68705-6_15
2017-01-01T00:00:00ZPolynomial bounds for chromatic number II: Excluding a star-forestScott, AlexSeymour, PaulSpirkl, Sophiehttp://hdl.handle.net/10012/185352022-08-12T02:31:29Z2022-10-01T00:00:00ZPolynomial bounds for chromatic number II: Excluding a star-forest
Scott, Alex; Seymour, Paul; Spirkl, Sophie
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 that, whenever such a statement holds, fH can be chosen to be a
polynomial. The Gyárfás–Sumner conjecture is only known to be true for a modest set of forests H,
and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known
when H is a five-vertex path. Here we prove Esperet's conjecture when each component of H is a
star.
This is the peer reviewed version of the following article: Scott, A., Seymour, P., & Spirkl, S. (2022). Polynomial bounds for chromatic number II: Excluding a star-forest. Journal of Graph Theory, 101(2), 318–322. https://doi.org/10.1002/jgt.22829, which has been published in final form at https://doi.org/10.1002/jgt.22829. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.
2022-10-01T00:00:00ZPolynomial bounds for chromatic number. III. Excluding a double starScott, AlexSeymour, PaulSpirkl, Sophiehttp://hdl.handle.net/10012/185342022-08-12T02:31:28Z2022-10-01T00:00:00ZPolynomial bounds for chromatic number. III. Excluding a double star
Scott, Alex; Seymour, Paul; Spirkl, Sophie
A “double star” is a tree with two internal vertices. It is known that the Gyárfás-Sumner conjecture
holds for double stars, that is, for every double star H, there is a function fH such that if G does
not contain H as an induced subgraph then x(G) ≤ fH(w(G)) (where x, w are the chromatic number
and the clique number of G). Here we prove that fH can be chosen to be a polynomial.
This is the peer reviewed version of the following article: Scott, A., Seymour, P., & Spirkl, S. (2022). Polynomial bounds for chromatic number. III. Excluding a double star. Journal of Graph Theory, 101(2), 323–340. https://doi.org/10.1002/jgt.22862, which has been published in final form at https://doi.org/10.1002/jgt.22862. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.
2022-10-01T00:00:00ZPiercing axis-parallel boxesChudnovsky, MariaSpirkl, SophieZerbib, Shirahttp://hdl.handle.net/10012/185332022-08-12T02:31:27Z2018-01-01T00:00:00ZPiercing axis-parallel boxes
Chudnovsky, Maria; Spirkl, Sophie; Zerbib, Shira
Let F be a finite family of axis-parallel boxes in Rd such that F contains no k + 1 pairwise disjoint boxes. We prove that if F contains a subfamily M of k pairwise disjoint boxes with the property that for every F E F and M E M with F ∩ M ≠ 6= Ø, either F contains a corner of M or M contains 2d-1 corners of F, then F can be pierced by O(k) points. One consequence of this result is that if d = 2 and the ratio between any of the side lengths of any box is bounded by a constant, then F can be pierced by O(k) points. We further show that if for each two intersecting boxes in F a corner of one is contained in the other, then F can be pierced by at
most O(k log log(k)) points, and in the special case where F contains only cubes this bound improves to O(k).
2018-01-01T00:00:00Z