Deleting vertex of graph G lowers the chromatic number by one

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I'm currently not really able to come up with a proof for this particular problem.




Let $G=(V,E)$ be a graph with the property that $∀ v ∈ V$, $χ(G−v) = χ(G)−1$. Show that $G$ is connected.




In other words: If you delete any vertex of the graph, the chromatic number will be reduced by one.



From my understanding this can only be the case in a complete graph. Since all vertices of a complete graph are connected with each other the chromatic number must be reduced by one if you delete any vertex. Which means that if I'm able to show that $G$ is a complete graph, I will have automatically proven that $G$ is connected since every complete graph is connected. I'm just not sure how to prove that $G$ is a complete graph or how to tackle this problem in general. I'm very thankful for any help I can get!




graph-theory coloring




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  • Note that odd cycle graphs also satisfy the given property...
    – Parcly Taxel
    Jul 16 at 17:02















up vote
1
down vote

favorite












I'm currently not really able to come up with a proof for this particular problem.




Let $G=(V,E)$ be a graph with the property that $∀ v ∈ V$, $χ(G−v) = χ(G)−1$. Show that $G$ is connected.




In other words: If you delete any vertex of the graph, the chromatic number will be reduced by one.



From my understanding this can only be the case in a complete graph. Since all vertices of a complete graph are connected with each other the chromatic number must be reduced by one if you delete any vertex. Which means that if I'm able to show that $G$ is a complete graph, I will have automatically proven that $G$ is connected since every complete graph is connected. I'm just not sure how to prove that $G$ is a complete graph or how to tackle this problem in general. I'm very thankful for any help I can get!




graph-theory coloring




share|cite|improve this question





















  • Note that odd cycle graphs also satisfy the given property...
    – Parcly Taxel
    Jul 16 at 17:02













up vote
1
down vote

favorite









up vote
1
down vote

favorite











I'm currently not really able to come up with a proof for this particular problem.




Let $G=(V,E)$ be a graph with the property that $∀ v ∈ V$, $χ(G−v) = χ(G)−1$. Show that $G$ is connected.




In other words: If you delete any vertex of the graph, the chromatic number will be reduced by one.



From my understanding this can only be the case in a complete graph. Since all vertices of a complete graph are connected with each other the chromatic number must be reduced by one if you delete any vertex. Which means that if I'm able to show that $G$ is a complete graph, I will have automatically proven that $G$ is connected since every complete graph is connected. I'm just not sure how to prove that $G$ is a complete graph or how to tackle this problem in general. I'm very thankful for any help I can get!




graph-theory coloring




share|cite|improve this question













I'm currently not really able to come up with a proof for this particular problem.




Let $G=(V,E)$ be a graph with the property that $∀ v ∈ V$, $χ(G−v) = χ(G)−1$. Show that $G$ is connected.




In other words: If you delete any vertex of the graph, the chromatic number will be reduced by one.



From my understanding this can only be the case in a complete graph. Since all vertices of a complete graph are connected with each other the chromatic number must be reduced by one if you delete any vertex. Which means that if I'm able to show that $G$ is a complete graph, I will have automatically proven that $G$ is connected since every complete graph is connected. I'm just not sure how to prove that $G$ is a complete graph or how to tackle this problem in general. I'm very thankful for any help I can get!





graph-theory coloring





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share|cite|improve this question




share|cite|improve this question








edited Jul 16 at 17:11
























asked Jul 16 at 16:36









Zelda_CompSci

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  • Note that odd cycle graphs also satisfy the given property...
    – Parcly Taxel
    Jul 16 at 17:02

















  • Note that odd cycle graphs also satisfy the given property...
    – Parcly Taxel
    Jul 16 at 17:02
















Note that odd cycle graphs also satisfy the given property...
– Parcly Taxel
Jul 16 at 17:02





Note that odd cycle graphs also satisfy the given property...
– Parcly Taxel
Jul 16 at 17:02











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Suppose $G$ is disconnected. The chromatic number of such a graph is the maximum of the chromatic numbers of its components, but a vertex can only belong to one component. Thus, if the removal of any particular vertex lowers the overall chromatic number, the component that vertex belongs to must have strictly higher chromatic number than that of the other components.



Repeating this argument, subjecting the remaining components to vertex removal, yields that every component's chromatic number is strictly higher than the others', which is absurd. Therefore, $G$ is connected.






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  • Thanks, I would've never solved this without your help.
    – Zelda_CompSci
    Jul 16 at 20:55










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1 Answer
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active

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes








up vote
1
down vote



accepted










Suppose $G$ is disconnected. The chromatic number of such a graph is the maximum of the chromatic numbers of its components, but a vertex can only belong to one component. Thus, if the removal of any particular vertex lowers the overall chromatic number, the component that vertex belongs to must have strictly higher chromatic number than that of the other components.



Repeating this argument, subjecting the remaining components to vertex removal, yields that every component's chromatic number is strictly higher than the others', which is absurd. Therefore, $G$ is connected.






share|cite|improve this answer





















  • Thanks, I would've never solved this without your help.
    – Zelda_CompSci
    Jul 16 at 20:55














up vote
1
down vote



accepted










Suppose $G$ is disconnected. The chromatic number of such a graph is the maximum of the chromatic numbers of its components, but a vertex can only belong to one component. Thus, if the removal of any particular vertex lowers the overall chromatic number, the component that vertex belongs to must have strictly higher chromatic number than that of the other components.



Repeating this argument, subjecting the remaining components to vertex removal, yields that every component's chromatic number is strictly higher than the others', which is absurd. Therefore, $G$ is connected.






share|cite|improve this answer





















  • Thanks, I would've never solved this without your help.
    – Zelda_CompSci
    Jul 16 at 20:55












up vote
1
down vote



accepted







up vote
1
down vote



accepted






Suppose $G$ is disconnected. The chromatic number of such a graph is the maximum of the chromatic numbers of its components, but a vertex can only belong to one component. Thus, if the removal of any particular vertex lowers the overall chromatic number, the component that vertex belongs to must have strictly higher chromatic number than that of the other components.



Repeating this argument, subjecting the remaining components to vertex removal, yields that every component's chromatic number is strictly higher than the others', which is absurd. Therefore, $G$ is connected.






share|cite|improve this answer













Suppose $G$ is disconnected. The chromatic number of such a graph is the maximum of the chromatic numbers of its components, but a vertex can only belong to one component. Thus, if the removal of any particular vertex lowers the overall chromatic number, the component that vertex belongs to must have strictly higher chromatic number than that of the other components.



Repeating this argument, subjecting the remaining components to vertex removal, yields that every component's chromatic number is strictly higher than the others', which is absurd. Therefore, $G$ is connected.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 16 at 17:00









Parcly Taxel

33.6k136588




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  • Thanks, I would've never solved this without your help.
    – Zelda_CompSci
    Jul 16 at 20:55
















  • Thanks, I would've never solved this without your help.
    – Zelda_CompSci
    Jul 16 at 20:55















Thanks, I would've never solved this without your help.
– Zelda_CompSci
Jul 16 at 20:55




Thanks, I would've never solved this without your help.
– Zelda_CompSci
Jul 16 at 20:55












 

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