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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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        Reference request: $n$-edge-coloring bipartite graph $K_{n,n}$ such that monochromatic parts are isomorphic

        I am finding references for the following problem: We call a $n\times n$ 0-1 matrix permutation if there are exactly one $1$ in each row/column. Suppose $A$ is a 0-1 matrix of size $n\times n$ in ...
        3
        votes
        1answer
        66 views

        Ear decompositions and spanning trees

        I am looking for a reference for the following theorem: Theorem: Let $G$ be a 2-connected, simple, undirected graph, and let $T$ be a spanning tree. Then $G$ has an ear decomposition in which every ...
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        votes
        1answer
        124 views

        Which groups contain a comb?

        The comb is the undirected simple graph with nodes $\mathbb{N} \times \mathbb{N}$ where $\mathbb{N} \ni 0$ and edges $$ \{\{(m,n), (m,n+1)\}, \{(m,0), (m+1,0)\} \;|\; m \in \mathbb{N}, n \in \mathbb{N}...
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        1answer
        54 views

        Contracting non-adjacent points in the icosahedron

        Are there $2$ non-adjacent points in the icosahedron graph $G$ such that contracting them leaves the Hadwiger number unchanged?
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        1answer
        117 views

        When can any graph $G$ be expressed as a union of $\alpha(G)$ complete graphs?

        If for any graph $H$ we define $\alpha(H)$ to be the cardinality of any maximum size indepedent set in $H$. Then under what conditions can any graph $G$ be expressed as a union of $\alpha(G)$ complete ...
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        Bipartite allocation with minimum cost

        Given two vertex sets $V_1$ and $V_2$. The vertices in $V_2$ have a limitation on the maximum degree of each vertex being $K$. I need to find an allocation algorithm such that every pair of vertices ...
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        1answer
        62 views

        Expected size of matchings in a cubic graph

        Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$? In other ...
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        80 views

        Find large “induced” bipartite graph in a dense graph?

        Do there exist constants $d>0$, $0<c<1$, $\delta>0$ so that for all large $n$, there exists a graph $H$ satisfying $$e_H\ge dn^2,$$ and then no matter how we remove some edges from $H$ to ...
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        vote
        0answers
        25 views

        Succinct circuits and NEXPTIME-complete problems

        I am fascinated by a recent fact I was reading: Succinct Circuits are simple machines used to descibe graphs in exponentially less space, which leads to the downside that solving a problem on that ...
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        votes
        0answers
        41 views

        Fastest Algorithm to calculate Graph pebbling number?

        I am interested in Graph Pebbling, and in particular what are the fastest known algorithm is to find the pebbling number of a graph. Also, i am interested whether there are lower limits on the runtime ...
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        Algorithms for Detecting the Completion of a Triangle in a Stream of Edges

        I need to efficiently determine in a complete weighted graph $G$ the sequence of triangles according to descending order of circumferences. My idea would be to incrementally construct a new graph $...
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        0answers
        59 views

        Are all even regular undirected Cayley graphs of Class 1?

        Are even order Cayley graphs of Class 1, that is, can they be edge-colored with exactly $m$ colors, where $m$ is the degree of each vertex? I think yes, because of the symmetry the Cayley graphs ...
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        96 views

        Which cubic graphs can be orthogonally embedded in $\mathbb R^3$?

        By an orthogonal embedding of a finite simple graph I mean an embedding in $\mathbb R^3$ such that each edge is parallel to one of the three axis. To avoid trivialities, let's require that (the ...
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        1answer
        67 views

        Maximizing “happy” vertices in splitting an infinite graph

        This question is motivated by a real life task (which is briefly described after the question.) Let $G=(V,E)$ be an infinite graph. For $v\in V$ let $N(v) = \{x\in V: \{v,x\}\in E\}$. If $S\subseteq ...
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        1answer
        52 views

        Figuring out a consistent definition for the percolation backbone

        In the context of percolation, e.g., bond/site percolation, random graph connectivity in 2-3 dimensions, etc., once the percolation threshold is reached, that is the system is spanned by an infinite ...

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