is_valid_degree_sequence_erdos_gallai

is_valid_degree_sequence_erdos_gallai(deg_sequence)

Returns True if deg_sequence can be realized by a simple graph.

The validation is done using the Erdős-Gallai theorem [EG1960].

Parameters:

deg_sequencelist

A list of integers

Returns:

validbool

True if deg_sequence is graphical and False if not.

Notes

This implementation uses an equivalent form of the Erdős-Gallai criterion. Worst-case run time is \(O(n)\) where \(n\) is the length of the sequence.

Specifically, a sequence d is graphical if and only if the sum of the sequence is even and for all strong indices k in the sequence,

\[\sum_{i=1}^{k} d_i \leq k(k-1) + \sum_{j=k+1}^{n} \min(d_i,k) = k(n-1) - ( k \sum_{j=0}^{k-1} n_j - \sum_{j=0}^{k-1} j n_j )\]

A strong index k is any index where d_k >= k and the value n_j is the number of occurrences of j in d. The maximal strong index is called the Durfee index.

This particular rearrangement comes from the proof of Theorem 3 in [2].

The ZZ condition says that for the sequence d if

\[|d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)}\]

then d is graphical. This was shown in Theorem 6 in [2].

References

[1]

A. Tripathi and S. Vijay. “A note on a theorem of Erdős & Gallai”, Discrete Mathematics, 265, pp. 417-420 (2003).

[2] (1,2)

I.E. Zverovich and V.E. Zverovich. “Contributions to the theory of graphic sequences”, Discrete Mathematics, 105, pp. 292-303 (1992).

[EG1960]

Erdős and Gallai, Mat. Lapok 11 264, 1960.