An official website of the United States government
Abstract 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 , there is a function such that if does not contain as an induced subgraph then (where are the chromatic number and the clique number of ). Here we prove that can be chosen to be a polynomial.
more »
« less
Polynomial bounds for chromatic number VII. Disjoint holes
Abstract Aholein a graph is an induced cycle of length at least four, and a ‐multiholein is the union of pairwise disjoint and nonneighbouring holes. It is well known that if does not contain any holes then its chromatic number is equal to its clique number. In this paper we show that, for any integer , if does not contain a ‐multihole, then its chromatic number is at most a polynomial function of its clique number. We show that the same result holds if we ask for all the holes to be odd or of length four; and if we ask for the holes to be longer than any fixed constant or of length four. This is part of a broader study of graph classes that are polynomially ‐bounded.
more »
« less
- PAR ID:
- 10413311
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Journal of Graph Theory
- Volume:
- 104
- Issue:
- 3
- ISSN:
- 0364-9024
- Page Range / eLocation ID:
- p. 499-515
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract The classical Hadwiger conjecture dating back to 1940s states that any graph of chromatic number at leastrhas the clique of orderras a minor. Hadwiger's conjecture is an example of a well‐studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph onnvertices of independence numberat mostr. If true Hadwiger's conjecture would imply the existence of a clique minor of order. Results of Kühn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition thatGisH‐free for some bipartite graphHthen one can find a polynomially larger clique minor. This has recently been extended to triangle‐free graphs by Dvořák and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graphH, answering a question of Dvořák and Yepremyan. In particular, we show that any‐free graph has a clique minor of order, for some constantdepending only ons. The exponent in this result is tight up to a constant factor in front of theterm.more » « less
-
Abstract The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Prałat noted that around the clique chromatic number of the random graph changes by when we increase the edge‐probability by , but left the details of this surprising “jump” phenomenon as an open problem. We settle this problem, that is, resolve the nature of this polynomial “jump” of the clique chromatic number of the random graph around edge‐probability . Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of up to logarithmic factors for any edge‐probability .more » « less
-
Aichholzer, Oswin; Wang, Haitao (Ed.)A graph is said to contain K_k (a clique of size k) as a weak immersion if it has k vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger’s conjecture: Every graph of chromatic number k contains K_k as a weak immersion. We prove this conjecture for graphs with at most 1.4(k-1) vertices. As an application, we make some progress on Albertson’s conjecture on crossing numbers of graphs, according to which every graph G with chromatic number k satisfies cr(G) ≥ cr(K_k). In particular, we show that the conjecture is true for all graphs of chromatic number k, provided that they have at most 1.4(k-1) vertices and k is sufficiently large.more » « less
-
We consider how to generate graphs of arbitrary size whose chromatic numbers can be chosen (or are well-bounded) for testing graph coloring algorithms on parallel computers. For the distance-1 graph coloring problem, we identify three classes of graphs with this property. The first is the Erdős-Rényi random graph with prescribed expected degree, where the chromatic number is known with high probability. It is also known that the Greedy algorithm colors this graph using at most twice the number of colors as the chromatic number. The second is a random geometric graph embedded in hyperbolic space where the size of the maximum clique provides a tight lower bound on the chromatic number. The third is a deterministic graph described by Mycielski, where the graph is recursively constructed such that its chromatic number is known and increases with graph size, although the size of the maximum clique remains two. For Jacobian estimation, we bound the distance-2 chromatic number of random bipartite graphs by considering its equivalence to distance-1 coloring of an intersection graph. We use a “balls and bins” probabilistic analysis to establish a lower bound and an upper bound on the distance-2 chromatic number. The regimes of graph sizes and probabilities that we consider are chosen to suit the Jacobian estimation problem, where the number of columns and rows are asymptotically nearly equal, and have number of nonzeros linearly related to the number of columns. Computationally we verify the theoretical predictions and show that the graphs are often be colored optimally by the serial and parallel Greedy algorithms.more » « less
