[Paper Review] Problems and memories

András Gyárfás|arXiv (Cornell University)|Jul 6, 2013

Philippine History and Culture26 references3 citations

TL;DR

This paper reflects on decades of mathematical collaboration with Paul Erdős, presenting key results and open problems in extremal graph theory, including cycle structures in sparse graphs, chordal subgraphs, monochromatic domination, and cycle partitioning in edge-colored complete graphs. Central contributions include conjectures on cycle coverage and chromatic edge-coloring, with partial results and improved bounds on Ramsey-type problems.

ABSTRACT

I state some open problems coming from joint work with Paul Erdős

Motivation & Objective

• To explore and resolve open problems in extremal graph theory inspired by Paul Erdős, particularly concerning sparse graphs and cycle structures.
• To investigate the existence and size of chordal subgraphs in dense graphs, especially near the threshold of $ n^2/3 $ edges.
• To examine monochromatic domination in 2- and 3-colored complete graphs, aiming to bound the number of vertices needed to dominate a large fraction of the vertex set in one color.
• To study the minimum number of vertex-disjoint monochromatic cycles needed to cover the vertex set of an $ r $-colored complete graph.
• To analyze the structure of nearly bipartite graphs and the conditions under which they can be made bipartite by removing few vertices.

Proposed method

• Analyzing extremal graph families such as $ G(n) $, the set of $ n $-vertex graphs with $ 2n-2 $ edges and no proper subgraph of minimum degree 3, to study cycle existence patterns.
• Using the concept of cross-intersecting families and Turán-type arguments to derive bounds on the minimum number of edges forcing a $ K_{k+1} $-free graph to contain a $ K_{k+1} $ upon edge addition.
• Applying probabilistic and extremal combinatorial techniques to bound the size of chordal subgraphs in graphs with $ rac{n^2}{3} $ or more edges.
• Employing Ramsey-theoretic methods and color-avoiding domination arguments to analyze monochromatic substructures in 2- and 3-colored complete graphs.
• Using the regularity lemma and structural graph decomposition to prove cycle partitioning results in $ r $-colored complete graphs.
• Introducing and analyzing the concept of $ (p,q) $-colorings of complete graphs, where every $ K_p $ spans at least $ q $ colors, and deriving bounds on the minimum number of colors $ f(n,p,q) $ required.

Experimental results

Research questions

• RQ1 Does every graph in $ G(n) $, with $ n $ vertices, $ 2n-2 $ edges, and no proper subgraph of minimum degree 3, contain cycles of all lengths from 3 up to some $ k $ that tends to infinity with $ n $?
• RQ2 Can every graph with $ n $ vertices and more than $ n^2/3 $ edges contain a chordal subgraph with at least $ 8n/3 - 4 $ edges?
• RQ3 In a 3-edge-colored complete graph $ K_n $, is there always a color in which at most 3 vertices dominate at least $ 2n/3 $ vertices?
• RQ4 Is the cycle partition number of an $ r $-colored complete graph at most $ r $?
• RQ5 Is $ f(n,5,9) $ linear in $ n $, i.e., is there a constant $ c $ such that $ K_n $ admits a proper edge coloring with $ cn $ colors where the union of any two color classes contains no path or cycle with four edges?

Key findings

• Conjecture 1 posits that every graph in $ G(n) $ contains cycles of all lengths from 3 to $ k $, where $ k o ofty $ as $ n o ofty $, though this remains unproven.
• It is shown that any graph with $ n $ vertices and at least $ n^2/3 $ edges contains a chordal subgraph with at least $ 2n - 3 $ edges, and this bound is tight for the complete tripartite graph.
• A weaker bound of $ 7n/3 - 6 $ edges is proven for chordal subgraphs in graphs with more than $ n^2/3 $ edges, improving upon earlier results.
• For 2-colored $ K_n $, it is proven that the vertex set can be covered by at most $ 2ig floorig floor $ monochromatic paths of the same color, and the conjecture that $ ig floorig floor $ suffice remains open.
• In 3-colored $ K_n $, it is shown that no more than 22 vertices dominate at least $ 2n/3 $ vertices in one color, and this bound has been improved to 4 by Kral et al., bringing it close to the conjectured bound of 3.
• For $ f(n,4,5) $, the bounds $ 5(n-1)/6 o f(n,4,5) o n $ are established, and the paper calls for tighter estimates, while $ f(n,5,9) $ is bounded below by $ 2n - 6 $ and above by $ 2n^{1 + c/ig floorig floor} $, with the question of linearity remaining open.