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pro vyhledávání: '"05c12"'
A $(1,1,2,2)$-coloring of a graph is a partition of its vertex set into four sets two of which are independent and the other two are $2$-packings. In this paper, we prove that every claw-free cubic graph admits a $(1,1,2,2)$-coloring. This implies th
Externí odkaz:
http://arxiv.org/abs/2409.15455
Given a finite set of red and blue points in $\mathbb{R}^d$, the MST-ratio is the combined length of the Euclidean minimum spanning trees of red points and of blue points divided by the length of the Euclidean minimum spanning tree of the union of th
Externí odkaz:
http://arxiv.org/abs/2409.11079
Let $\mathscr{B}_n = \{ \pm x_1, \pm x_2, \pm x_3, \cdots, \pm x_{n-1}, x_n \}$ where $n>1$ is fixed, $x_i \in \mathbb{R}^+$, $i = 1, 2, 3, \cdots, n$ and $x_1 < x_2 < x_3 < \cdots < x_n$. Let $\phi(\mathscr{B}_n)$ be the set of all non-empty subsets
Externí odkaz:
http://arxiv.org/abs/2409.09317
Autor:
Cochran, Garner, Wang, Zhiyu
Erd\H{o}s, Pach, Pollack, and Tuza [J. Combin. Theory Ser. B, 47(1) (1989), 73--79] proved that the diameter of a connected $n$-vertex graph with minimum degree $\delta$ is at most $\frac{3n}{\delta+1}+O(1)$. The oriented diameter of an undirected gr
Externí odkaz:
http://arxiv.org/abs/2409.06587
Autor:
Asif, Tauseef, Haidar, Ghulam, Yousafzai, Faisal, Khan, Murad Ul Islam, Khan, Qaisar, Fatima, Rakea
A resolving set for a simple graph $G$ is a subset of vertex set of $G$ such that it distinguishes all vertices of $G$ using the shortest distance from this subset. This subset is a metric basis if it is the smallest set with this property. A resolvi
Externí odkaz:
http://arxiv.org/abs/2409.12199
We study Babai numbers and Babai $k$-spectra of paths and cycles. We completely determine the Babai numbers of paths $P_n$ for $n>1$ and $1 \leq k \leq n-1$, and the Babai $k$-spectra for $P_n$ when $1 \leq k \leq n/2$. We also completely determine B
Externí odkaz:
http://arxiv.org/abs/2409.04869
We study the problem of calculating noncommutative distances on graphs, using techniques from linear algebra, specifically, Birkhoff-James orthogonality. A complete characterization of the solutions is obtained in the case when the underlying graph i
Externí odkaz:
http://arxiv.org/abs/2409.04146
Autor:
Di Pietro, Gabriele, Ripà, Marco
For each pair of positive integers $(a,b)$ such that $a \geq 0$ and $b > 1$, the present paper provides a necessary and sufficient condition for the existence of Hamiltonian cycles visiting all the vertices of any $k$-dimensional grid $\{0,1\}^k \sub
Externí odkaz:
http://arxiv.org/abs/2409.03073
In this paper, we study the oriented diameter of power graphs of groups. We show that a $2$-edge connected power graph of a finite group has oriented diameter at most $4$. We prove that the power graph of a cyclic group of order $n$ has oriented diam
Externí odkaz:
http://arxiv.org/abs/2409.02457
Autor:
Erickson, William Q., Herden, Daniel, Meddaugh, Jonathan, Sepanski, Mark R., Minyard, Mitchell, Rosengartner, Kyle
Given a finite simple graph $G$, let $\operatorname{M}(G)$ denote its 2-distance graph, in which two vertices are adjacent if and only if they have distance 2 in $G$. In this paper, we consider the periodic behavior of the sequence $G, \operatorname{
Externí odkaz:
http://arxiv.org/abs/2409.02306