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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
        36 views

        Partitioning vertex set to maximize weights of inter-class edges?

        An interesting problem has come up in my work, and I haven't quite been able to find references to it so I thought I'd ask here. Suppose we have some complete, weighted graph with vertex set $V$. Is ...
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        vote
        0answers
        25 views

        Minimum transitive dominating subtournament

        For any positive integer $k$, does there exist a tournament such that the smallest dominating set that forms a transitive subtournament has size exactly $k$? A tournament that does not work for $k>...
        1
        vote
        1answer
        43 views

        Maximum number of perfect matchings in a graph of genus $g$ balanced $k$-partite graph

        What is the maximum number of perfect matchings a genus $g$ balanced $k$-partite graph (number of vertices for each color in all possible $k$-colorings is within a difference of $1$) can have? I am ...
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        vote
        0answers
        47 views

        What are the various kinds of graphs that can be defined on $C(X)$

        I was considering the space $C(X)$ where $X$ is a topological space and $C(X)$ is the set of all continuous functions from $X$ to $\Bbb R$. What are the various kinds of graphs that can be defined on ...
        4
        votes
        1answer
        40 views

        Calculate number of vertices adjacent to a clique, but not each other

        I have a network of N sensors and a test that tells me whether two sensor outputs are definitely not causally related. This allows me to construct a causality graph where each sensor is a vertex and ...
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        votes
        0answers
        15 views

        Difference between Adjacent strong edge coloring and vertex distinguishing strong edge coloring

        What is the exact difference between the adjacent strong edge coloring and vertex distinguishing proper edge coloring of graphs? This paper refers to the fact that the adjacent strong edge coloring of ...
        4
        votes
        1answer
        117 views

        Integers with a Hamiltonian Square Path

        Let $n>1$ be an integer and set $[n]=\{1,\ldots,n\}$. We say that $n$ has a "Hamiltonian Square Path" if there is a bijection $\varphi:[n]\to[n]$ such that for all $k\in [n-1]$ we have that $\...
        3
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        0answers
        35 views

        Explicit, small resolving sets for Hamming graphs

        Definition. Let $G = (V;E)$ be a finite, undirected graph. $R = \{r_1, \ldots, r_k \} \subseteq V$. $R$ resolves $G$ if $$ V \to [0, \infty]^k, v \mapsto (d_G(v,r_1), \ldots, d_G(v, r_k)) $$ is ...
        2
        votes
        1answer
        61 views

        Diagonal shortcuts to minimize all-pairs shortest-paths in grid graph

        Augment the grid graph $G$ on lattice points $[1,n]^2$, which connects each point to its four distance-$1$ vertical and horizontal neighbors. Augment $G$ to $G'$ by adding in one of the two $\sqrt{2}$ ...
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        vote
        0answers
        12 views

        Can a bramble of maximal order be efficiently found from a tree decomposition of minimal width?

        Let $G$ be a connected graph with vertices $V(G)$. A bramble of $G$ is a set of connected subgraphs $H_1,\ldots,H_n$ such that for each $i$ and $j$, $H_i$ touches $H_j$; that is, either $H_i$ ...
        6
        votes
        1answer
        137 views

        Strict unfriendly partitions

        Given a graph $G$, an unfriendly partition of $G$ is a partition of $V(G)$ into two classes, such that for every class, every vertex has at least as many neighbors in the other class as in its own ...
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        votes
        0answers
        112 views

        A question about dominating circuits in cubic graphs

        Let $G$ be a 3-connected cubic graph with a dominating circuit $C$, that is, a circuit such that all edges in $G$ have at least one endvertex in $C$. Let $D$ be another circuit and let the symmetric ...
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        vote
        1answer
        48 views

        Regularization of arbitrary graphs

        Let $G=(V,E)$ be a finite, simple, undirected graph. Let $\Delta(G)$ be the maximal degree of $G$. Is there a finite graph $G'=(V', E')$ with the following properties? $V\subseteq V'$, $E \subseteq E'...
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        votes
        0answers
        60 views

        Non-isomorphic graphs with same Tutte polynomial [closed]

        I've been looking for some non-isomorphic graphs with the same Tutte polynomial. I'm aware of this thread and this thread, however my understanding of matroids is non-existent, and they are a bit ...
        2
        votes
        1answer
        93 views

        Edge colorability and Hamiltonicity of certain classes of cubic graphs (MO graphs)

        Let $G$ be a simple cubic graph (that is, 3-regular). A dominating circuit of $G$ is a circuit $C$ such that each edge of $G$ has an endvertex in $C$. The circuit $C$ is chordless if no edge which is ...

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