A note about monochromatic components in graphs of large minimum degree
نویسندگان
چکیده
For all positive integers $r\geq 3$ and $n$ such that $r^2-r$ divides an affine plane of order $r$ exists, we construct $r$-edge colored graph with minimum degree $(1-\frac{r-2}{r^2-r})n-2$ the largest monochromatic component has less than $\frac{n}{r-1}$. This generalizes example Guggiari Scott and, independently, Rahimi for $r=3$ thus disproves a conjecture Gyarfas Sarkozy exists.
منابع مشابه
Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition
It is well-known that in every k-coloring of the edges of the complete graph Kn there is a monochromatic connected component of order at least n k−1 . In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For k = 2 the authors proved that δ(G) > 3n 4 ensures a monochromatic connected component with at least δ(G) + 1 vertices in every...
متن کاملH-Free Graphs of Large Minimum Degree
We prove the following extension of an old result of Andrásfai, Erdős and Sós. For every fixed graph H with chromatic number r+1 ≥ 3, and for every fixed > 0, there are n0 = n0(H, ) and ρ = ρ(H) > 0, such that the following holds. Let G be an H-free graph on n > n0 vertices with minimum degree at least ( 1 − 1 r−1/3 + ) n. Then one can delete at most n2−ρ edges to make G r-colorable.
متن کاملA Note on Independent Sets in Graphs with Large Minimum Degree and Small Cliques
Graphs with large minimum degree containing no copy of a clique on r vertices (Kr) must contain relatively large independent sets. A classical result of Andrásfai, Erdős, and Sós implies that Kr-free graphs G with degree larger than ((3r−7)/(3r− 4))|V (G)| must be (r− 1)-partite. An obvious consequence of this result is that the same degree threshold implies an independent set of order (1/(r − ...
متن کاملA note on the vertex-distinguishing proper coloring of graphs with large minimum degree
We prove that the number of colors required to color properly the edges of a graph of order n and δ(G) > n/3 in such a way that any two vertices are incident with different sets of colors is at most ∆(G) + 5.
متن کاملDiameter Two Graphs of Minimum Order with Given Degree Set
The degree set of a graph is the set of its degrees. Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. Furthermore, the minimum order of such a graph is determined. A graph is 2-self- centered if its radius and diameter are two. In this paper for ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Discussiones Mathematicae Graph Theory
سال: 2023
ISSN: ['1234-3099', '2083-5892']
DOI: https://doi.org/10.7151/dmgt.2390