GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
89 views

A vertex colouring of a graph $G=(V, E)$ with $k$ coulours is a mapping $c: V \rightarrow \{1, \dots , k\}$ such that $c(u) \neq c(v)$ for every $(u, v) \in E$. Consider the following statements:

  1. If every vertex in $G$ has degree at most $d$ then $G$ admits a vertex coulouring using $d+1$ colours.
  2. Every cycle admits a vertex colouring using 2 colours
  3. Every tree admits a vertex colouring using 2 colours

Which of the above statements is/are TRUE? Choose from the following options:

  1. only i
  2. only i and ii
  3. only i and iii
  4. only ii and iii
  5. i, ii, and iii
asked in Graph Theory by Veteran (76k points)   | 89 views

1 Answer

+2 votes
Best answer
i is true, since in worst case the graph can be complete. So, d+1 colours are necessary for graph containing vertices with degree atmost 'd' .

ii is false since cyles with odd no of vertices require 3 colours.

iii is true, since each level of the tree must be coloured in an alternate fashion. We can do this with two colours.

Therefore, option c is correct.
answered by Active (1k points)  
selected by


Top Users Mar 2017
  1. rude

    4272 Points

  2. sh!va

    2994 Points

  3. Rahul Jain25

    2804 Points

  4. Kapil

    2608 Points

  5. Debashish Deka

    2244 Points

  6. 2018

    1414 Points

  7. Vignesh Sekar

    1338 Points

  8. Akriti sood

    1246 Points

  9. Bikram

    1246 Points

  10. Sanjay Sharma

    1016 Points

Monthly Topper: Rs. 500 gift card

21,452 questions
26,771 answers
60,972 comments
22,985 users