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Results tagged with trees
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user 384338
A tree is a connected graph without cycles, with a finite or infinite number of vertices. There are many variants of trees, according to further constraints or decorations.
3
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0
answers
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If the girth of a $2k$-regular graph $G$ is larger than the diameter of a tree $T$ with $k$ ...
I want to prove that ‘If the girth of a $2k$-regular graph $G$ is larger than the diameter of a $k$-edge tree $T$, then $G$ is covered by edge-disjoint copies of $T$.’
I tried several ways to solve th …
- 341
1
vote
Two independent spanning trees of $2$-connected graph
By hypothesis, $G'$ has two spanning trees $T_1'$ and $T_2'$ made by $E$, containing two independent $uv'$-paths for any $v' \in V(G')-\{u\}$. … If $P_k$ is an edge, $T_1'$ and $T_2'$ are the desired trees. So assume $V(P_k) \geq 3$. Label the vertices in $P_k$ by $y_1,\cdots,y_n$ in clockwise direction. …
- 341
2
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2
answers
527
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Two independent spanning trees of $2$-connected graph
Then $G$ has two spanning trees such that for every vertex $v$, the $u,v$-paths in the trees are independent.
I tried to show this, but surprisingly, I have proved another statement. …
- 341
2
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1
answer
383
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Maximum number of leaf blocks in 3-regular (cubic) graph
The definition of block is
Block of $G$ is a maximal subgraph $G'$ of $G$ with no cut vertex of $G'$ itself.
Of course, there can exist many blocks in $G$.
In particular, isolated vertices, edges in …
- 341