# Questions tagged [regular-graph]

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### How to construct 4-regular graphs with few Hamiltonian decompositions?

A Hamiltonian decomposition of a finite simple graph is a partition of its edge set so that each partition class forms a Hamiltonian cycle. This is only possible if the graph is $2k$-regular.
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2
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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 ...

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### If the girth of a $2k$-regular graph $G$ is larger than the diameter of a tree $T$ with $k$ edges, then $G$ is decomposed into copies of $T$

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 ...

1
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1
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### Is every $k$-edge connected $k$-regular graph Hamiltonian?

A graph $G$ is Hamiltonian if there is a Hamiltonian cycle in $G$.
Suppose $G$ is a $k$-edge connected $k$-regular graph with $k>1$.
Does this ensure that $G$ is Hamiltonian?
If not, how about ...

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### Smallest $3$-regular graph with a unique perfect matching

What is the smallest 3-regular graph to have a unique perfect matching?
With a large enough number of nodes, it is possible for a 3-regular graph to have no perfect matching (example can be seen in ...

2
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2
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### Efficiently generating all regular/bidegreed graphs

There is a related question on how to generate all regular graphs; however, the procedure is random and repeats the generated graphs. Plus, there is no stop condition, unless recording the total ...

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### Chromatic number of regular graphs using spectra

There exist inequalities relating the maximum and minimum eigenvalues of the adjacency matrix of a graph with its chromatic numbers, i.e. the Wilf's and Hoffmann's inequalities, which put together ...

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### Decomposition of regular graphs

Let $G$ be a regular simple graph with degree $\Delta=n-k-1$ and order $m$. Let $C_k$ be the regular graph which is formed by removing a $k$-factor from the complete graph $K_{n}$. I think we could ...

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### Lower bound for $\vert \det A \vert $ for the adjacency matrix of regular graphs

Assume $G$ is a simple $k$-regular graph of order $n$ with adjacency matrix $A$ which is non-singular.
Does anyone know some lower bounds for $\vert \det (A) \vert$ with respect to $n$, $k$ or both?
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### The diameter of random regular graphs

In 1982, B. Bollobas and Vega in the paper gave the configurational model to generate $r$-regular random graphs. They gave the following theorem (Theorem 1 in the paper).
Theorem: Let $r\geq 3$ and $\...