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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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        1answer
        58 views

        Perfect graphs condition could be weakened?

        The perfect graphs are generally defined as those graphs whose every induced subgraph has its chromatic number equal to its clique number. Now,are there some examples where the clique number of graph ...
        1
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        0answers
        81 views

        Chromatic number of certain graphs with high maximum degree

        Let $G$ be the graph of even order $n$ and size $\ge\frac{n^2}{4}$ which is a Cayley graph on a nilpotent group but not complete. Can the chromatic number of this graph be determined in polynomial ...
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        0answers
        464 views

        Is this representation of Go (game) irreducible?

        This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph. Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...
        2
        votes
        1answer
        66 views

        Define a homomorphism of a set of graphs to its power set

        Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G_1$ and $G_2$ is, $G_1\cup G_2$ $=\langle V(G_1)\cup V(G_2), (E(G_1)\...
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        0answers
        51 views

        Combinatorial equation system with exponentially many equations in quadratic many variables

        A certain question on graph theory (about the existance of graphs with a certain coloring inherited by perfect matchings) can be translated into the satisfiability problem of a certain set of ...
        2
        votes
        0answers
        177 views

        Is there a known proof that $R(5,5)\leq 47$ in Ramsey theory?

        As an application to a model describing graphs with partial information, I found what might be an (as yet unverified) proof that $R(5,5)\leq 47$. According to the Dynamic Survey of Ramsey Numbers at ...
        3
        votes
        1answer
        47 views

        Connected hypergraphs

        We say that a hypergraph $H=(V,E)$ is connected if the following condition holds: for all $S\subseteq V$ with $\emptyset\neq S \neq V$ there is $e\in E$ that meets both $S$ and $V\setminus S$, i.e. ...
        6
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        0answers
        105 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 ...
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        0answers
        81 views

        Digraphs with same number of semiwalks

        This is a follow-up question to Characterisation of walk-equivalent digraphs. Question: Do there exists two directed graphs $G$ and $H$ consisting of the same number ($n$) of vertices, such that \...
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        votes
        0answers
        35 views

        Generalization of Menger's Theorem to Infinite Graphs

        Aharoni and Berger generalized Menger's Theorem to infinite graphs: For any digraph, and any subsets A and B, there is a family F of disjoint paths from A to B and a set separating B from A consisting ...
        1
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        0answers
        25 views

        Worst case performance of heuristic for the non-eulerian Windy Postman Problem

        The Windy Postman Problem seeks the cheapest tour in a complete undirected graph, that traverses each edge at least once; the cost of traversing an edge is positive and may depend on the direction, in ...
        1
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        0answers
        22 views

        Definition of k-partite hypergraph

        I would like to know the standard definition of k-partite hypergraph. There are two natural generalizations of k-partite graph to k-partite hypergraph: 1. For all edges e, any two vertices in e are ...
        15
        votes
        7answers
        972 views

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

        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 "...
        1
        vote
        1answer
        88 views

        Characterisation of walk-equivalent digraphs

        Setting Let $G=(V,E)$ be an undirected graph. A walk $\pi$ in $G$ of length $k$ is a sequence of $k+1$ vertices $v_1,\ldots,v_{k+1}$ such that for each $i\in[1,k]$, $\{v_i,v_{i+1}\}\in E$. Let $H=(W,F)...
        0
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        0answers
        38 views

        Graphs “weak” in context of cutting subgraphs

        Lately I've been looking into graphs (simple, undirected, finite) that are in some way weak when it comes to connectivity, that is: Let $G$ be a graph of order $n$. We'll say that $G$ is $k$-weak if ...

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