380 research outputs found
Large generalized books are p-good
An r-book of size q is a union of q (r+1)-cliques sharing a common r-clique.
We find exactly the Ramsey number of a p-clique versus r-books of sufficiently
large size. Furthermore, we find asymptotically the Ramsey number of any fixed
p-chromatic graph versus r-books of sufficiently large size. The key element in
our proofs is Szemeredi's Regularity Lemma.Comment: 16 pages, accepted in JCT
Construction of near-optimal vertex clique covering for real-world networks
We propose a method based on combining a constructive and a bounding heuristic to solve the vertex clique covering problem (CCP), where the aim is to partition the vertices of a graph into the smallest number of classes, which induce cliques. Searching for the solution to CCP is highly motivated by analysis of social and other real-world networks, applications in graph mining, as well as by the fact that CCP is one of the classical NP-hard problems. Combining the construction and the bounding heuristic helped us not only to find high-quality clique coverings but also to determine that in the domain of real-world networks, many of the obtained solutions are optimal, while the rest of them are near-optimal. In addition, the method has a polynomial time complexity and shows much promise for its practical use. Experimental results are presented for a fairly representative benchmark of real-world data. Our test graphs include extracts of web-based social networks, including some very large ones, several well-known graphs from network science, as well as coappearance networks of literary works' characters from the DIMACS graph coloring benchmark. We also present results for synthetic pseudorandom graphs structured according to the Erdös-Renyi model and Leighton's model
A note on circular chromatic number of graphs with large girth and similar problems
In this short note, we extend the result of Galluccio, Goddyn, and Hell,
which states that graphs of large girth excluding a minor are nearly bipartite.
We also prove a similar result for the oriented chromatic number, from which
follows in particular that graphs of large girth excluding a minor have
oriented chromatic number at most , and for the th chromatic number
, from which follows in particular that graphs of large girth
excluding a minor have
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The Chromatic Structure of Dense Graphs
This thesis focusses on extremal graph theory, the study of how local constraints on a graph affect its macroscopic structure. We primarily consider the chromatic structure: whether a graph has or is close to having some (low) chromatic number.
Chapter 2 is the slight exception. We consider an induced version of the classical Turán problem. Introduced by Loh, Tait, Timmons, and Zhou, the induced Turán number ex(n, {H, F-ind}) is the greatest number of edges in an n-vertex graph with no copy of H and no induced copy of F. We asymptotically determine ex(n, {H, F-ind}) for H not bipartite and F neither an independent set nor a complete bipartite graph. We also improve the upper bound for ex(n, {H, K_{2, t}-ind}) as well as the lower bound for the clique number of graphs that have some fixed edge density and no induced K_{2, t}.
The next three chapters form the heart of the thesis. Chapters 3 and 4 consider the Erdős-Simonovits question for locally r-colourable graphs: what are the structure and chromatic number of graphs with large minimum degree and where every neighbourhood is r-colourable? Chapter 3 deals with the locally bipartite case and Chapter 4 with the general case.
While the subject of Chapters 3 and 4 is a natural local to global colouring question, it is also essential for determining the minimum degree stability of H-free graphs, the focus of Chapter 5. Given a graph H of chromatic number r + 1, this asks for the minimum degree that guarantees that an H-free graph is close to r-partite. This is analogous to the classical edge stability of Erdős and Simonovits. We also consider the question for the family of graphs to which H is not homomorphic, showing that it has the same answer.
Chapter 6 considers sparse analogues of the results of Chapters 3 to 5 obtaining the thresholds at which the sparse problem degenerates away from the dense one.
Finally, Chapter 7 considers a chromatic Ramsey problem first posed by Erdős: what is the greatest chromatic number of a triangle-free graph on vertices or with m edges? We improve the best known bounds and obtain tight (up to a constant factor) bounds for the list chromatic number, answering a question of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot
Recent developments in graph Ramsey theory
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress
Fractional colorings of partial -trees with no large clique
Dvo\v{r}\'ak and Kawarabayashi [European Journal of Combinatorics, 2017]
asked, what is the largest chromatic number attainable by a graph of treewidth
with no subgraph? In this paper, we consider the fractional version
of this question. We prove that if has treewidth and clique number , then , and we
show that this bound is tight for . We also show that for each
value , there exists a graph of a large treewidth
and clique number satisfying .Comment: 9 page
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