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Consider the alternating group graph, here defined as a Cayley graph on the alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\ldots,(1,2,n),(1,n,2),\ldots,(1,4,2),(1,3,2)\}$. Note that when $n=3$, the graph reduces to a triangle.

Observing that the clique size is just $3$(this can be seen by observing the structure around the identity), I propose that the graph is properly $3$-colorable for all $n$. But, can this be proved? I suppose using left cosets of $A_3$ with respect to $A_n$ would help us here. In fact, it helped in getting $3$-coloring for $A_4$. Whereas, I am struck even for $A_5$. The SageMath software gives me the expected result up to $A_6$. Any hints? Thanks beforehand.

Consider the alternating group graph, here defined as a Cayley graph on the alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\dotsc,(1,2,n),(1,n,2),\dotsc,(1,4,2),(1,3,2)\}$. Note that when $n=3$, the graph reduces to a triangle.

Observing that the clique size is just $3$  (this can be seen by observing the structure around the identity), I propose that the graph is properly $3$-colorable for all $n$. But, can this be proved? I suppose using left cosets of $A_3$ with respect to $A_n$ would help us here. In fact, it helped in getting $3$-coloring for $A_4$. Whereas, I am struck even for $A_5$. The SageMath software gives me the expected result up to $A_6$. Any hints?

Consider the alternating group graph, here defined as a Cayley graph using on the alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\ldots,(1,2,n),(1,n,2),\ldots,(1,4,2),(1,3,2)\}$. Note that when $n=3$, the graph reduces to a triangle.

Observing that the clique size is just $3$(this can be seen by observing the structure around the identity), I propose that the graph is properly $3$-colorable for all $n$. But, can this be proved? I suppose using left cosets of $A_3$ with respect to $A_n$ would help us here. In fact, it helped in getting $3$-coloring for $A_4$. Whereas, I am struck even for $A_5$. The SageMath software gives me the expected result up to $A_6$. Any hints? Thanks beforehand.

Consider the alternating group graph, here defined as a Cayley graph on the alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\ldots,(1,2,n),(1,n,2),\ldots,(1,4,2),(1,3,2)\}$. Note that when $n=3$, the graph reduces to a triangle.

Observing that the clique size is just $3$(this can be seen by observing the structure around the identity), I propose that the graph is properly $3$-colorable for all $n$. But, can this be proved? I suppose using left cosets of $A_3$ with respect to $A_n$ would help us here. In fact, it helped in getting $3$-coloring for $A_4$. Whereas, I am struck even for $A_5$. The SageMath software gives me the expected result up to $A_6$. Any hints? Thanks beforehand.

Consider the alternating group graph, here defined as a Cayley graph using on the Alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\ldots,(1,2,n),(1,n,2),\ldots,(1,4,2),(1,3,2)\}$. Note that when $n=3$, the graph reduces to a triangle.

Observing that the clique size is just $3$(this can be seen by observing the structure around the identity), I propose that the graph is properly $3$-colorable for all $n$. But, can this be proved? I suppose using left cosets of $A_3$ with respect to $A_n$ would help us here. In fact, it helped in getting $3$-coloring for $A_4$. Whereas, I am struck even for $A_5$. The SageMath software gives me the expected result upto $A_6$. Any hints? Thanks beforehand.

Consider the alternating group graph, here defined as a Cayley graph using on the alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\ldots,(1,2,n),(1,n,2),\ldots,(1,4,2),(1,3,2)\}$. Note that when $n=3$, the graph reduces to a triangle.

Observing that the clique size is just $3$(this can be seen by observing the structure around the identity), I propose that the graph is properly $3$-colorable for all $n$. But, can this be proved? I suppose using left cosets of $A_3$ with respect to $A_n$ would help us here. In fact, it helped in getting $3$-coloring for $A_4$. Whereas, I am struck even for $A_5$. The SageMath software gives me the expected result up to $A_6$. Any hints? Thanks beforehand.