It is proved that all 4-edge-colorings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree Δ = 3 is a threshold for Kempe equivalence of (Δ+1)-edge-colorings, as such an equivalence does not hold in general when Δ = 4. One extra color allows a similar result in this latter case; however, namely, when Δ≤4 it is shown that all (Δ+2)-edge-colorings are Kempe equivalent. © 2011 Wiley Periodicals, Inc. J Graph Theory

(Bojan Mohar is on leave from IMFM 8 FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. © 2012 Wiley Periodicals, Inc.)

/pdf/kempe-equivalence-of-edge-colorings-2lprizxc1b.pdf

Kempe Equivalence of Edge-Colorings

/pdf/on-a-conjecture-of-mohar-concerning-kempe-equivalence-of-1lnuk3jcbe.pdf

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

https://scholarworks.smith.edu/cgi/viewcontent.cgi?article=1000&context=mth_facpubs

Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs

Some results on Reed's Conjecture about ω,Δ, and χ with respect to α

Tension-continuous (shortly TT) mappings are mappings between the edge sets of graphs. They generalize graph homomorphisms. At the same time, tension-continuous mappings are a dual notion to flow-continuous mappings, and the context of nowhere-zero flows motivates several questions considered in this paper. Extending our earlier research we define new constructions and operations for graphs (such as graphs @D"M(G)) and give evidence for the complex relationship of homomorphisms and TT mappings. Particularly, solving an open problem, we display pairs of TT-comparable and homomorphism-incomparable graphs with arbitrarily high connectivity. We give a new (and more direct) proof of density of TT order and study graphs such that TT mappings and homomorphisms from them coincide; we call such graphs homotens. We show that most graphs are homotens, on the other hand every vertex of a nontrivial homotens graph is contained in a triangle. This provides a justification for our construction of homotens graphs.

On tension-continuous mappings

Claude Berge - Sculptor of Graph Theory.- ?-path-connectivity and mk-generation: an Upper Bound on m.- Automated Results and Conjectures on Average Distance in Graphs.- Brambles, Prisms and Grids.- Dead Cell Analysis in Hex and the Shannon Game.- Ratios of Some Domination Parameters in Graphs and Claw-free Graphs.- Excessive Factorizations of Regular Graphs.- Odd Pairs of Cliques.- Recognition of Perfect Circular-arc Graphs.- On Edge-maps whose Inverse Preserves Flows or Tensions.- On the Extremal Number of Edges in 2-Factor Hamiltonian Graphs.- Generalized Colourings (Matrix Partitions) of Cographs.- A Note on [k, l]-sparse Graphs.- Even Pairs in Bull-reducible Graphs.- Kernels in Orientations of Pretransitive Orientable Graphs.- Nonrepetitive Graph Coloring.- A Characterization of the 1-well-covered Graphs with no 4-cycles.- A Graph-theoretical Generalization of Berge's Analogue of the Erd?s-Ko-Rado Theorem.- Independence Polynomials and the Unimodality Conjecture for Very Well-covered, Quasi-regularizable, and Perfect Graphs.- Precoloring Extension on Chordal Graphs.- On the Enumeration of Bipartite Minimum Edge Colorings.- Kempe Equivalence of Colorings.- Acyclic 4-choosability of Planar Graphs with Girth at Least 5.- Automorphism Groups of Circulant Graphs - a Survey.- Hypo-matchings in Directed Graphs.- On Reed's Conjecture about ?,? and ?.- On the Generalization of the Matroid Parity Problem.- Reconstruction of a Rank 3 Oriented Matroids from its Rank 2 Signed Circuits.- The Normal Graph Conjecture is True for Circulants.- Two-arc Transitive Near-polygonal Graphs.- Open Problems.

Graph theory in Paris : proceedings of a conference in memory of Claude Berge

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Graph theory in Paris : proceedings of a conference in memory of Claude Berge