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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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        44 views

        Planar graphs with perfect matching count in linear time?

        We can find Pfaffian orientation and take determinant to compute permanent in $O(n^\omega)$ time where $\omega$ is exponent of matrix multiplication. We know that permanent of $O(n)$ vertex planar ...
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        1answer
        117 views

        For what graph does the following algebraic property hold?

        Let $G=(V,E)$ be a simple graph. My question: For what graph $G$, does there exist a permutation $\sigma$ on $V$ such that $$\prod_{uv\in E}(x_{\sigma(u)}-x_{\sigma(v)})=-\prod_{uv\in E}(x_u-x_v)?$$ ...
        1
        vote
        1answer
        100 views

        Bookthickness of covering space

        A book embedding of a graph G consists of placing the vertices of G on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The page number is a ...
        7
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        1answer
        141 views

        Co-spectral fractional isomorphic graphs with different Laplacian spectrum

        I am looking for two undirected graphs $G$ and $H$ of the same order (i.e., they have the same number of vertices) such that $G$ and $H$ are cospectral (i.e., their adjacency matrices $A_G$ and $A_H$ ...
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        1answer
        58 views

        A vertex transitive graph has a near perfect/ matching missing an independent set of vertices

        Consider a power of cycle graph $C_n^k\,\,,\frac{n}{2}>k\ge2$, represented as a Cayley graph with generating set $\{1,2,\ldots, k,n-k,\ldots,n-1\}$ on the Group $\mathbb{Z}_n$. Supposing I remove ...
        1
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        1answer
        69 views

        Shortest path on graphs

        I would like to now if there has been any work on related problems, that is, shortest path problem in dynamically evolving graphs.
        2
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        0answers
        80 views

        Can entropy of a network be written as a polynomial?

        In my research, I met a problem here. Consider a weighted graph Laplacian matrix $$\mathcal{L}_w(\mathcal{G}) = DWD^T,$$ where $D$ is a incidence matrix and $W$ is diagonal with each diagonal ...
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        19 views

        Sufficient Condition for the Existence of Vertex Disjoint Shortest Paths

        Let $G(V,E)$ be a symmetric connected graph with $n$ vertices and let $D\in\mathbb{N}_0^{n\times n}$ be the matrix containing as entries $d_{uv}$ the least number of edges on a path from $u$ to $v$. ...
        1
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        1answer
        55 views

        Finding an element of the homology group of a graph which is everywhere nonzero

        Let $\Gamma$ be an oriented graph (multiple edges between vertices are allowed), and let $G$ be a finite abelian group. We define $H_1(\Gamma,G)$ to be the set of $G$-linear combinations of edges of $\...
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        1answer
        116 views

        Highly asymmetric regular graph

        Let $G$ be a regular connected simple graph on $n$ vertices with chromatic number $\chi$ and maximum degree $\Delta$. Then, it is implied that $G$ is $\chi$-partite. Suppose, we remove one of the ...
        5
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        1answer
        79 views

        Existence of regular factors in dense graphs

        All graphs here are finite and simple. A $d$-factor of a graph is a spanning regular subgraph of degree $d$. Where can I find theorems of this nature, for constants $a,b,c\gt 0$: If $G$ is a graph ...
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        0answers
        49 views

        Finding fundamental Kempe-locking configurations

        Let $xy$ be an edge in a planar triangulation $T$ that is at least 4-connected and let $uxvy$ be the 4-cycle delineating the 4-face of the near-triangulation $G_{xy}$ obtained by deleting $xy$ from $T$...
        3
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        1answer
        67 views

        Equitable edge coloring of graphs

        Consider a simple regular graph $G$ with $n$ vertices and $E$ edges. Then, can we say that the edges can be colored equitably in $\Delta+1$ colors? By equitability is meant that a proper $\Delta+1$ ...
        5
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        0answers
        36 views

        Normal colorings of bridgeless cubic graphs

        Definition (informal) A normal edge-5-coloring of a bridgeless cubic graph $G$ is a proper 5 coloring of the edges of the graph, so that for each edge $e\in E(G)$, either $e$ and the four edges ...
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        51 views

        Helly vs Strong p-Helly Property of Hypergraphs

        I am not clear about the difference between Helly and Strong p-Helly property. For example hypergraph H(V, E), V = { 1,2,3 } and E = {(1,2), (2,3), (1,3)} has non-empty set for each pair of ...

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