- FindStat

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Identifier

St001357: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

([(0,1)],2) => 1

([],3) => 0

([(1,2)],3) => 0

([(0,2),(1,2)],3) => 0

([(0,1),(0,2),(1,2)],3) => 2

([],4) => 0

([(2,3)],4) => 0

([(1,3),(2,3)],4) => 0

([(0,3),(1,3),(2,3)],4) => 0

([(0,3),(1,2)],4) => 1

([(0,3),(1,2),(2,3)],4) => 1

([(1,2),(1,3),(2,3)],4) => 0

([(0,3),(1,2),(1,3),(2,3)],4) => 1

([(0,2),(0,3),(1,2),(1,3)],4) => 2

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3

([],5) => 0

([(3,4)],5) => 0

([(2,4),(3,4)],5) => 0

([(1,4),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,4),(3,4)],5) => 0

([(1,4),(2,3)],5) => 0

([(1,4),(2,3),(3,4)],5) => 0

([(0,1),(2,4),(3,4)],5) => 0

([(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,3),(3,4)],5) => 0

([(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 0

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,3),(2,3),(2,4)],5) => 0

([(0,1),(2,3),(2,4),(3,4)],5) => 0

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 0

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 2

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([],6) => 0

([(4,5)],6) => 0

([(3,5),(4,5)],6) => 0

([(2,5),(3,5),(4,5)],6) => 0

([(1,5),(2,5),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0

([(2,5),(3,4)],6) => 0

([(2,5),(3,4),(4,5)],6) => 0

([(1,2),(3,5),(4,5)],6) => 0

([(3,4),(3,5),(4,5)],6) => 0

([(1,5),(2,5),(3,4),(4,5)],6) => 0

([(0,1),(2,5),(3,5),(4,5)],6) => 0

([(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 0

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(2,4),(2,5),(3,4),(3,5)],6) => 0

([(0,5),(1,5),(2,4),(3,4)],6) => 0

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 0

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 0

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 0

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 0

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,4),(2,3)],6) => 1

([(1,5),(2,4),(3,4),(3,5)],6) => 0

([(0,1),(2,5),(3,4),(4,5)],6) => 1

([(1,2),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 0

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 1

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 0

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 0

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 1

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 1

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 3 + q^{2}$

$F_{4} = 5 + 3\ q + 2\ q^{2} + q^{3}$

$F_{5} = 26 + 7\ q^{2} + q^{4}$

$F_{6} = 54 + 50\ q + 36\ q^{2} + 12\ q^{3} + 3\ q^{4} + q^{5}$

Description

The maximal degree of a regular spanning subgraph of a graph.
A

$k$ -regular spanning subgraph of a graph is also known as

$k$ -factor. This statistic is the largest

$k$ such that the graph has a

$k$ -factor.

References

[1] wikipedia:Graph_factorization

Code

def statistic(G):
    for d in range(max(G.degree()), 0, -1):
        H = find_regular_subgraph_degree(G.canonical_label().copy(immutable=True), d)
        if H:
            return H.degree()[0]
    return 0

@cached_function
def find_regular_subgraph_degree(G, d):
    if G.is_regular(d):
        return G
    if min(G.degree()) < d:
        return

    for u, v in G.edges(labels=False):
        if G.degree(u) > d and G.degree(v) > d:
            H = G.copy(immutable=False)
            H.delete_edge(u, v)
            H1 = find_regular_subgraph_degree(H.canonical_label().copy(immutable=True), d)
            if H1:
                return H1

Created

Feb 25, 2019 at 08:52 by Martin Rubey

Updated

Dec 26, 2020 at 02:37 by Martin Rubey