On Induced Colourful Paths in Triangle-free Graphs

@article{Babu2017OnIC,
  title={On Induced Colourful Paths in Triangle-free Graphs},
  author={Jasine Babu and Manu Basavaraju and L. Sunil Chandran and Mathew C. Francis},
  journal={Electron. Notes Discret. Math.},
  year={2017},
  volume={61},
  pages={69-75}
}
Given a graph G = (V, E) whose vertices have been properly coloured, we say that a path in G is colourful if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy Theorem that every properly coloured graph contains a colourful path on χ(G) vertices. It is interesting to think of what analogous result one could obtain if one considers induced colourful paths instead of just colourful paths. We explore a conjecture that states that every properly coloured triangle… Expand
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References

SHOWING 1-10 OF 10 REFERENCES
Induced Colorful Trees and Paths in Large Chromatic Graphs
TLDR
Here the following weaker result is proved providing some evidence towards the conjecture that in every proper coloring of a-chromatic triangle-free graph there is an induced colorful path P_k. Expand
Colorful induced subgraphs
TLDR
It is proved that there exists a function f(k, n) such that for any colored graph G, if χ(G)>f(ω(G), n) then G induces either a colorful out directed star with n leaves or a colorful directed path on n vertices. Expand
A Generalization of the Gallai–Roy Theorem
  • Hao Li
  • Mathematics, Computer Science
  • Graphs Comb.
  • 2001
TLDR
For graphs, a positive answer to the following question of Fajtlowicz is given: if G is a graph with chromatic number χ(G), then for any proper coloring of G of χ (G) colors and for any vertex v∈V(G, there is a path P starting at v which represents all χ(_) colors. Expand
Rainbow Paths with Prescribed Ends
TLDR
It is proved that every connected graph with at least one edge has a proper $k-coloring (for some $k$) such that every vertex of color $i$ has a neighbor of color$i+1$ (mod $k) along each edge. Expand
Induced Subgraphs of Graphs with Large Chromatic Number IX: Rainbow Paths
TLDR
It is proved that for all nonnegative integers k,s there exists c with the following property: for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. Expand
Introduction to Graph Theory
1. Fundamental Concepts. What Is a Graph? Paths, Cycles, and Trails. Vertex Degrees and Counting. Directed Graphs. 2. Trees and Distance. Basic Properties. Spanning Trees and Enumeration.Expand
Induced subtrees in graphs of large chromatic number
Our paper proves special cases of the following conjecture: for any fixed tree T there exists a natural number f = f (T) to that every triangle-free graph of chromaticnumber f(T) contains T as anExpand
Graph Theory
TLDR
This book provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal, and is suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science. Expand
Graph theory, Graduate texts
  • 2007
Problems from the world surrounding perfect graphs