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      All Questions

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      6
      votes
      0answers
      84 views

      Squared squares and partitions of $K_{nn}$

      This is inspired by a recent question. Define a square square sum (SSS) of order $n$ to be any partition $$n^2=\sum_1^tc_ii^2 \tag{*}$$ of $n^2$ into square summands. Call it perfect if all $c_i \leq ...
      13
      votes
      7answers
      835 views

      Examples of proofs by making reduction to a finite set [on hold]

      This is a very abstract question, I hope this is appropriate. Suppose $T$ is some claim over some infinite set $A$, for example, let $A$ be the set of all loopless planar graphs, and $T$ be the claim "...
      3
      votes
      1answer
      151 views

      Diameter of Cayley graphs of finite simple groups

      Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article). THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
      11
      votes
      4answers
      358 views

      A specific collection of subgraphs in $K_{70, 70}$

      Does there exist a collection of subgraphs $\{\Gamma_i\}_{i = 1}^{24}$ of $K_{70, 70}$, that satisfy the following two properties: 1)$\Gamma_i \cong K_{i, i} \forall 1 \leq i \leq 24$; 2)Any ...
      4
      votes
      0answers
      48 views

      Dinitz Conjecture extension to rectangles

      The Dinitz Conjecture, which was proved later in a more general form by Galvin, stated that given an $n\times n$ array, its elements could be filled exactly like a latin square, where the elements in ...
      -5
      votes
      0answers
      81 views

      Are $\sqrt{1 + x}$ and $\log(1 + x)$ limit distribution of some special random walks? [on hold]

      We know that $\exp(-x^2)$ is the limit distribution of a binomial random walk. Are $\sqrt{1 + x}$ and $\log(1 + x)$ also limit distribution of some nature random walks ? Maybe some "restricted" ...
      2
      votes
      1answer
      59 views

      List coloring of tripartite graph

      Let $G$ be a tripartite graph with partite sets $A,B,C$. The graphs $A\cup B$, $B\cup C$ and $C\cup A$ are each bipartite. Let the maximum degree of the graph be $\Delta$. Now, we know that the ...
      0
      votes
      1answer
      65 views

      If the core of a graph is a forest, then it is Class 1

      It is a standard result, due to Fournier, that if the core of a graph (the induced graph by the vertices having their degrees equal to maximum degree of the graph) is a forest (acyclic), then the ...
      7
      votes
      0answers
      156 views

      Matrix of high rank mod $2$: must it have a large non-singular minor (with disjoint rows and columns)?

      Let $A$ be a $2n$-by-$2n$ matrix with entries in $\mathbb{Z}/2\mathbb{Z}$ such that, for every $2n$-by-$2n$ diagonal matrix $D$ with entries in $\mathbb{Z}/2\mathbb{Z}$, the matrix $A+D$ has rank $\...
      1
      vote
      1answer
      123 views

      Quotient graph of a tree

      We know that every graph is isomorphic to a subgraph of a complete graph. Similarly, can we say that every graph is isomorphic to a quotient graph of a tree?
      0
      votes
      1answer
      54 views

      Combining three matchings to form a maximal matching

      Consider a regular tripartite graph $G$ with maximum degree $\Delta\ge3$ and parts $A,B,C$. Now, the induced subgraphs $A\cup B, B\cup C$ and $A\cup C$ are all bipartite. Now, is there a way to ...
      1
      vote
      1answer
      100 views

      Number formation and bridged graphs, connection or coincidence?

      Bridged graphs sequence $g(n) =$ "Number of simple connected bridged graphs on $n+2$ nodes". We have $g(n)=1, 3, 10, 52, 351, 3714,\dots$ from A052446. Number formation sequence We also have $f(n) ...
      1
      vote
      1answer
      50 views

      The Total Graph is similar to a line graph

      Consider the total graph of a regular graph. From the structure, it seems that it has a similar structure to the line graph ( two different sub-cliques joining at a single point) except that, in ...
      2
      votes
      0answers
      63 views

      Zero-One law for Hamiltonian path subgraphs of Hamming Distance Graphs?

      $(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}...
      0
      votes
      1answer
      51 views

      Chromatic Polynomial when two disjoint graphs are joined at $2$ distinct points [closed]

      Consider a graph with chromatic polynomial $P(x)$ joined to a clique of order $k$ in two distinct points (joining here just means interesection of points). Then, what is the chromatic polynomial of ...

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