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- W4280517242 abstract "Let $G$ be a graph, let $m$ be a positive integer, and let $f$ be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $Ssubseteq V(G)$, $$Omega_m(Gsetminus S)le sum_{vin S}big(f(v)-2mbig)+m+Omega_m(G[S]),$$ then $G$ has a spanning $m$-tree-connected subgraph $H$ containing an arbitrary given subgraph with maximum degree at most $m$ such that for each vertex $v$, $d_H(v)le f(v)$, where $G[S]$ denotes the induced subgraph of $G$ with the vertex set $S$ and $Omega_m(G_0)$ is a parameter to measure $m$-tree-connectivity of a given graph $G_0$. By applying this result, we prove that every $k$-edge-connected graph $G$ has a spanning $m$-tree-connected subgraph $H$ such that for each vertex $v$, $d_H(v)le biglceil frac{m}{k}(d_G(v)-2m)bigrceil+2m$, where $kge 2m$; also if $G$ is $k$-tree-connected, then $H$ can be found such that for each vertex $v$, $d_H(v)le biglceil frac{m}{k}(d_G(v)-m)bigrceil+m$, where $kge m$. Next, we prove that if for all $Ssubseteq V(G)$, $$frac{1}{2}Omega_2(Gsetminus S)le sum_{vin S}(f(v)-3/2)+1,$$ then $G$ admits a spanning closed trail meeting each vertex $v$ at most $f(v)$ times. From this result, we conclude that every $4$-tough graph of order at least three admits a spanning closed trail meeting each vertex at most twice. This result confirms a weaker version of a long-standing conjecture due to Chv'atal (1973)." @default.
- W4280517242 created "2022-05-22" @default.
- W4280517242 creator A5090095075 @default.
- W4280517242 date "2022-05-10" @default.
- W4280517242 modified "2023-09-23" @default.
- W4280517242 title "Spanning tree-connected subgraphs and spanning closed trails with small degrees" @default.
- W4280517242 doi "https://doi.org/10.48550/arxiv.2205.05044" @default.
- W4280517242 hasPublicationYear "2022" @default.
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