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99-Graph:Is there a graph with 99 vertices in which every edge(i.e. pair of joined vertices) belong to a unique triangle and every nonedge(pair of unjoined vertices) to a unique quadrilateral?

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To guarantee the "every" and "unique" conditions is not easy. – Rupei Xu Apr 6 '14 at 7:40
You of course have $\binom{99}{2} = 3t + 4q$ where $t$ is the number of triangles, and $q$ is the number of quadrilaterials, so there are some restrictions, if such a solution exists. – Per Alexandersson Apr 6 '14 at 8:49
Professor John Conway (Princeton University) would like to offer $1,000 for this problem to the one who first solves it. – Rupei Xu Apr 9 '14 at 6:21
up vote 15 down vote accepted

First we will prove the graph is regular.

Let $x,y$ be two non-adjacent vertices, and let $a,b$ be their common neighbours. Define $X$ to be the neighbourhood of $x$ other than $a,b$, and $Y$ to be the neighbourhood of $y$ other than $a,b$.

Considering the edge $ax$, there is a unique vertex $u\in X$ adjacent to both of them. Considering the non-edge $yu$, there must be exactly one edge from $u$ to $Y$. Similarly for the common neighbourbour of $b$ and $x$. For a vertex in $v\in X$ not adjacent to $a$ or $b$, the two common neighbours of $v$ and $y$ must lie in $Y$.

Consider the bipartite graph with parts $X,Y$ and the edges between them. We have proved that each part has 2 vertices of degree 1 and the others of degree 2. This is only possible if $|X|=|Y|$, which proves that $x$ and $y$ have the same degree. This proves the graph is regular. A simple count shows the degree must be 14.

Now you are looking for a strongly-regular graph of order 99, degree 14, $\lambda=1$ and $\mu=2$. According to Andries Brouwer's table, the existence is unknown.

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Can you say why a and b are unique? (I'm thinking of a red-blue complete graph where the red edges form triangles and the blue edges form quadrilaterals. Maybe that is the wrong picture?) – The Masked Avenger Apr 6 '14 at 15:46
@The Masked Avenger: If I understand the question correctly, any two non-adjacent vertices lie on a unique quadrilateral, which means they have exactly two common neighbours. There are no edge colours. – Brendan McKay Apr 6 '14 at 16:00
Also, for x and y not connected, I am getting that the sum of their red degrees is at least 90. Am I doing something wrong? – The Masked Avenger Apr 6 '14 at 16:10
I see. I thought the edge was part of the (blue) quadrilateral. Certainly your view is more intriguing. Thanks. – The Masked Avenger Apr 6 '14 at 16:12
Thank you for your helpful analysis and references. Life is so beautiful with a nice Professor like you! – Rupei Xu Apr 9 '14 at 6:20

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