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Introduction to Graph Theory
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
A graph is a regular graph if the degree of every vertex is the same integer. A k-regular graph is one for which the degree of each vertex is k. A 3-regular graph is often called a cubic graph. The complete graph Kn $ K_n $ is an (n-1) $ (n-1) $ -regular graph.
Flows and colorings
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Delia Garijo, Andrew Goodall, Jaroslav Nešetřil
Jaeger [645] used the fact, due to Tutte and to C. Nash-Williams, that a 4-edge-connected graph has two edge-disjoint spanning trees in order to prove that every 4-edge-connected graph has a nowhere-zero 4-flow. As nowhere-zero 4-flows of a cubic graph correspond to proper edge 3-colorings, the 4-flow conjecture implies that every bridgeless cubic graph without a Petersen minor is 3-edge-colorable. The latter was another conjecture of Tutte. A proof was announced by N. Robertson, P. Seymour, and R. Thomas in [966], thereby giving a strengthening of the four color theorem, which is equivalent to the assertion that planar cubic graphs without a bridge are 3-edge-colorable.
A New Design Chart Method of Journal Bearings Based on a Simplified Thermohydrodynamic Lubrication Theory
Published in Tribology Transactions, 2020
Thus, it looks promising that each of and is determined respectively by giving the values of the two newly defined coordinate variables with and incorporated. As a result, each of and needs to be expressed in a cubic graph. However, when each of and is shown on a 2D plane, a 2D contour map needs to be used. Consequently, the next step is determining the isotherms of traced on a new design chart by combining Figs. 3a and 5a and also determining the isoviscous lines of traced on another new design chart by combining Figs. 3b and 5b.