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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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      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 ...
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      47 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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      0answers
      31 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 ...
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      24 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 ...
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      16 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 ...
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      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 "...
      1
      vote
      1answer
      83 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)...
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      17 views

      What's the best method for sorting many multiple-characteristic items into multiple-characteristic bins? [on hold]

      ? There are 100 items to be sorted into just over 100 bins, where no bin has more than one item and not every bin is necessarily filled, but every item must be placed. ? There are certain items that ...
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      0answers
      33 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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      vote
      0answers
      24 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
      204 views
      +50

      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
      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
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      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 ...
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      0answers
      23 views

      cycle structure of bounded genus graphs

      Given fixed genus $k$ is there some $q=f(k)$ for which there are $q$ sets of cycles $S_1, S_2,.., S_q$ each cycle of $S_i$ uses only verticies of $V_i \subset V$ each $G[V_i]$ is planar $S_i$ is a ...

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