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        Questions tagged [graph-theory]

        Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.

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        0answers
        31 views

        Treewidth related properties of a bipartite graph with bounded local crossing number and diameter

        If a bipartite degree at most $3$ graph on $O(n^2)$ vertices with diameter at most $O(\log n)$ has property that every edge intersects at most $O(\log n)$ edges on a planar drawing then does any of ...
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        0answers
        220 views

        Why do we assume that $\mathcal{A}$ is an algebra in this 2003 paper of Bobkov and Tetali?

        In the following paper (extended version here), at the beginning of section 3, the authors give two axioms about $\mathcal{A}$. Axiom 1 is about $\mathcal{A}$ being an algebra. I do not see where this ...
        3
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        1answer
        156 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 $S$ ...
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        4answers
        452 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
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        0answers
        77 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 ...
        2
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        1answer
        77 views

        List coloring of tripartite graph [closed]

        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 ...
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        0answers
        9 views

        Iterated Inverse structures: polynomial representation of integer partitioning of preimages in Sigma Matrices (reference request)

        I am studying iterated preimage structures of functions on a finite set. The main structure of interest to me, the Sigma Matrix, is derived from a matrix listing the element-wise preimage sets at ...
        0
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        1answer
        71 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 ...
        1
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        1answer
        46 views

        Number of occurrences of subgraphs as a unique identifier

        Given $q \in \mathbb{N}$, let $B_q$ be a sequence of all (non isomorphic) connected graphs with at most $q$ vertices. Now for a given connected graph $G$, lets define signature of $G$ ($sig_q(G)$) as ...
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        0answers
        161 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
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        0answers
        60 views

        Is there a well-posed definition of game on a graph? Or a well defined category of games on graphs?

        All I ever found about this were natural language rules à la Asimov's three laws of robotics. The questions are straightforward questions: 1) Is there a well-posed mathematical definition of game on ...
        1
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        0answers
        29 views

        Determining the minimum weight maximal oriented subgraph of a complete directed graph

        Let $G(V,A,W):\ |V| = n,\ A=V\times V\setminus \lbrace (v,\ v)\rbrace,\ W\in\mathbb{R}_+^{n\times n},\ W^T\ne W $ be a complete directed graph with asymmetric weights. Questions: What is ...
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        0answers
        58 views

        Infimums of Poset of Unlabelled Subtrees

        I will use $T$ to refer to the set of unlabelled, rooted trees, and use $(t,r)$ to denote a tree and its root. Let $(T, \preceq)$ be a poset where $(t_1,r_1) \preceq (t_2,r_2)$ means that $t_1$ is a ...
        2
        votes
        1answer
        77 views

        Two cospectral (normal) digraphs which are not orthogonal similar

        Preliminaries A complex matrix $A$ is normal when $A$ and $A^*$ commute. A real matrix $A$ is normal when $A$ and $A^t$ commute. Two complex matrices $A$ and $B$ are said to be unitary similar if ...
        4
        votes
        3answers
        230 views

        Is there a name for this “stack” of graphs?

        Let $G_1,\ldots,G_m$ be a sequence of graphs, all having the same number $n$ of vertices. For each pair $(G_i, G_{i+1})$ we add $n$ edges that connect the vertices of $G_i$ and $G_{i+1}$ bijectively. ...

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