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        All Questions

        1
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        0answers
        26 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>...
        4
        votes
        1answer
        42 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 ...
        0
        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
        118 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 $\...
        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 ...
        5
        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 ...
        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 ...
        2
        votes
        1answer
        148 views

        History of algebraic graph theory

        I need a source about the history of algebraic graph theory. I mean for solving which problems or responding to what needs it was created? Indeed, I want to write a note about the history of the ...
        5
        votes
        1answer
        125 views

        Large dominating sets in tournaments

        It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $\lceil \log_2 n\rceil$. (See Fact 2.5 here.) What are tournaments such that any dominating set ...
        4
        votes
        1answer
        139 views

        Edge coloring a cycle plus triangles graph and a stronger problem

        Let $G$ be a simple “cycle plus triangles” graph, that is, a graph with $3k$ vertices, $k>1$, the edges of which can be partitioned into a set that induces a $3k$-circuit, together with sets that ...
        1
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        1answer
        46 views

        Injective edge choice functions in linear hypergraphs

        A linear hypergraph is a hypergraph $H=(V,E)$ such that for $e\in E$ we have $|e|\geq 2$, and if $e\neq e_1\in E$, then $|e\cap e_1| \leq 1$. An injective edge choice function of a linear ...
        2
        votes
        1answer
        47 views

        Number of distinct perfect matchings/near perfect matchings in an induced subgraph

        Consider a Class 1 graph with degree $\Delta\ge3$ and the induced subgraph formed by deleting a set of independent vertices of cardinality $\left\lfloor\frac{n}{\Delta}\right\rfloor$. Then, what is ...
        6
        votes
        1answer
        93 views

        Restricted independent set of the cycle graph $C_{3n}$

        Let $V$ be the vertices of the cycle graph $C_{3n}$. Suppose there is a partition of $V$ into sets of $3$, i.e. $V=\cup_{k=1}^{n}{V_k}$ where $|V_k|=3$ for $k$ in $1..n$. QUESTION: Is it possible ...
        0
        votes
        1answer
        60 views

        All even order graphs with $\Delta\ge\frac{n}{2}$ is Class 1

        Are all even order graphs with maximum degree $\ge\frac{|V(G)|}{2}$ Class 1(edge-colorable(chromatic index) with $\delta(G)$ colors)? Here, $|V(G)|$ detnotes the number of vertices in the graph. I ...
        5
        votes
        3answers
        152 views

        Disjunction number of a graph

        Let $S\neq \emptyset$ be a set. We make its powerset ${\cal P}(S)$ into a simple, undirected graph by saying that $A, B\in{\cal P}(S)$ form an edge if and only if $A\cap B=\emptyset$. The ...

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        山西福彩快乐十分钟
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