Understanding CLIQUE structure

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I am working on the following problem:



Recall the definition of a complete graph Kn is a graph with n vertices such that every vertex
is connected to every other vertex. Recall also that a clique is a complete subset of some graph.
The graph coloring problem consists of assigning a color to each of the vertices of a graph such
that adjacent vertices have different colors and the total number of colors used is minimized. We
define the chromatic number of a graph G to be this minimum number of colors required to color
graph G. Prove that the chromatic number of a graph G is no less than the size of a maximal
clique of G.



So far, I have been thinking about the problem and came up with the following:



  • In a clique, every vertex is adjacent to every other vertex

  • Therefore, since the problem states adjacent verteces always have different colors, the number of colors needed would be the number of verteces in the clique

  • Thus, the chromatic number of any graph G has to be at least the maximal size of the clique, per the definition of a clique.

Can someone verify if my understanding is correct? Or am I glaringly not understanding a clique?






algorithms graph-theory clique







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    up vote
    2
    down vote

    favorite












    I am working on the following problem:



    Recall the definition of a complete graph Kn is a graph with n vertices such that every vertex
    is connected to every other vertex. Recall also that a clique is a complete subset of some graph.
    The graph coloring problem consists of assigning a color to each of the vertices of a graph such
    that adjacent vertices have different colors and the total number of colors used is minimized. We
    define the chromatic number of a graph G to be this minimum number of colors required to color
    graph G. Prove that the chromatic number of a graph G is no less than the size of a maximal
    clique of G.



    So far, I have been thinking about the problem and came up with the following:



    • In a clique, every vertex is adjacent to every other vertex

    • Therefore, since the problem states adjacent verteces always have different colors, the number of colors needed would be the number of verteces in the clique

    • Thus, the chromatic number of any graph G has to be at least the maximal size of the clique, per the definition of a clique.

    Can someone verify if my understanding is correct? Or am I glaringly not understanding a clique?






    algorithms graph-theory clique







    share|cite|improve this question























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I am working on the following problem:



      Recall the definition of a complete graph Kn is a graph with n vertices such that every vertex
      is connected to every other vertex. Recall also that a clique is a complete subset of some graph.
      The graph coloring problem consists of assigning a color to each of the vertices of a graph such
      that adjacent vertices have different colors and the total number of colors used is minimized. We
      define the chromatic number of a graph G to be this minimum number of colors required to color
      graph G. Prove that the chromatic number of a graph G is no less than the size of a maximal
      clique of G.



      So far, I have been thinking about the problem and came up with the following:



      • In a clique, every vertex is adjacent to every other vertex

      • Therefore, since the problem states adjacent verteces always have different colors, the number of colors needed would be the number of verteces in the clique

      • Thus, the chromatic number of any graph G has to be at least the maximal size of the clique, per the definition of a clique.

      Can someone verify if my understanding is correct? Or am I glaringly not understanding a clique?






      algorithms graph-theory clique







      share|cite|improve this question













      I am working on the following problem:



      Recall the definition of a complete graph Kn is a graph with n vertices such that every vertex
      is connected to every other vertex. Recall also that a clique is a complete subset of some graph.
      The graph coloring problem consists of assigning a color to each of the vertices of a graph such
      that adjacent vertices have different colors and the total number of colors used is minimized. We
      define the chromatic number of a graph G to be this minimum number of colors required to color
      graph G. Prove that the chromatic number of a graph G is no less than the size of a maximal
      clique of G.



      So far, I have been thinking about the problem and came up with the following:



      • In a clique, every vertex is adjacent to every other vertex

      • Therefore, since the problem states adjacent verteces always have different colors, the number of colors needed would be the number of verteces in the clique

      • Thus, the chromatic number of any graph G has to be at least the maximal size of the clique, per the definition of a clique.

      Can someone verify if my understanding is correct? Or am I glaringly not understanding a clique?






      algorithms graph-theory clique




      algorithms graph-theory clique






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      asked 4 hours ago









      Jerry M.

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          Your understanding is right. So in particular, for any graph $G$ with a clique of size $k$, you will need at least $k$ colors to properly color it. In case this is not obvious, you can think about it via a contradiction. So assume $G$ contains a clique of size $k$, and you have a proper coloring with less than $k$ colors. It follows that the clique contains at least two vertices with the same color. By definition of a clique, these two vertices are adjacent. But this is a contradiction to the coloring being proper.



          Further, one might also wonder if the chromatic number of a graph would always equal the clique number. In this direction, you can have a look at perfect graphs for which the chromatic number of every induced subgraph equals its clique number. On the other hand, the Mycielski graphs show that there are graph that don't even contain a triangle (equivalently, a clique on 3 vertices), while the chromatic number can be arbitrarily large.






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            1 Answer
            1






            active

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            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            2
            down vote



            accepted










            Your understanding is right. So in particular, for any graph $G$ with a clique of size $k$, you will need at least $k$ colors to properly color it. In case this is not obvious, you can think about it via a contradiction. So assume $G$ contains a clique of size $k$, and you have a proper coloring with less than $k$ colors. It follows that the clique contains at least two vertices with the same color. By definition of a clique, these two vertices are adjacent. But this is a contradiction to the coloring being proper.



            Further, one might also wonder if the chromatic number of a graph would always equal the clique number. In this direction, you can have a look at perfect graphs for which the chromatic number of every induced subgraph equals its clique number. On the other hand, the Mycielski graphs show that there are graph that don't even contain a triangle (equivalently, a clique on 3 vertices), while the chromatic number can be arbitrarily large.






            share|cite|improve this answer
























              up vote
              2
              down vote



              accepted










              Your understanding is right. So in particular, for any graph $G$ with a clique of size $k$, you will need at least $k$ colors to properly color it. In case this is not obvious, you can think about it via a contradiction. So assume $G$ contains a clique of size $k$, and you have a proper coloring with less than $k$ colors. It follows that the clique contains at least two vertices with the same color. By definition of a clique, these two vertices are adjacent. But this is a contradiction to the coloring being proper.



              Further, one might also wonder if the chromatic number of a graph would always equal the clique number. In this direction, you can have a look at perfect graphs for which the chromatic number of every induced subgraph equals its clique number. On the other hand, the Mycielski graphs show that there are graph that don't even contain a triangle (equivalently, a clique on 3 vertices), while the chromatic number can be arbitrarily large.






              share|cite|improve this answer






















                up vote
                2
                down vote



                accepted







                up vote
                2
                down vote



                accepted






                Your understanding is right. So in particular, for any graph $G$ with a clique of size $k$, you will need at least $k$ colors to properly color it. In case this is not obvious, you can think about it via a contradiction. So assume $G$ contains a clique of size $k$, and you have a proper coloring with less than $k$ colors. It follows that the clique contains at least two vertices with the same color. By definition of a clique, these two vertices are adjacent. But this is a contradiction to the coloring being proper.



                Further, one might also wonder if the chromatic number of a graph would always equal the clique number. In this direction, you can have a look at perfect graphs for which the chromatic number of every induced subgraph equals its clique number. On the other hand, the Mycielski graphs show that there are graph that don't even contain a triangle (equivalently, a clique on 3 vertices), while the chromatic number can be arbitrarily large.






                share|cite|improve this answer












                Your understanding is right. So in particular, for any graph $G$ with a clique of size $k$, you will need at least $k$ colors to properly color it. In case this is not obvious, you can think about it via a contradiction. So assume $G$ contains a clique of size $k$, and you have a proper coloring with less than $k$ colors. It follows that the clique contains at least two vertices with the same color. By definition of a clique, these two vertices are adjacent. But this is a contradiction to the coloring being proper.



                Further, one might also wonder if the chromatic number of a graph would always equal the clique number. In this direction, you can have a look at perfect graphs for which the chromatic number of every induced subgraph equals its clique number. On the other hand, the Mycielski graphs show that there are graph that don't even contain a triangle (equivalently, a clique on 3 vertices), while the chromatic number can be arbitrarily large.







                share|cite|improve this answer












                share|cite|improve this answer



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                answered 3 hours ago









                Juho

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