What question did this study set out to answer?

To determine conditions under which a graph has a 2-factor containing cycles through specified vertices.

April 4, 2026Open Access

Degree-Sum Conditions for Graphs to Have 2-Factors with Cycles Through Specified Vertices

Key Points

To determine conditions under which a graph has a 2-factor containing cycles through specified vertices.
Analyzed graphs of order n with minimum degree at least k + 1.
Established conditions based on the degree sums of distinct nonadjacent vertices.
Applied combinatorial arguments to demonstrate existence of cycles.
Identified that the minimum sum of degrees condition leads to the existence of required cycles.
Showed that for specified vertices, at least k cycles can be formed in the 2-factor.
Verified that k distinct vertices can be part of the cycles D1, D2, ..., Dk.

Abstract

Let k 2 and n 1 be integers, let G be a graph of order n with minimum degree at least k + 1.Let v1, v2, , v k be k distinct vertices of G, and suppose that there existSuppose further that the minimum value of the sum of the degrees of two nonadjacent distinct vertices is greater than or equal to n + k-4 3 .Under these assumptions, we show that there is a 2-factor of G with k cycles D1, D2, , D k such that vi V (Di) for each 1 i k.

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