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This paper is an exploration of simple four-regular graphs in the plane (i.e. loop-free and with no more than one edge between any two nodes). Such graphs are fundamental to the theory of knots and links in three dimensional space, and their planar diagrams. We dedicate this paper to Frank Harary (1921–2005), whose fascination with graphs of knots inspired this work, and with whom we had the pleasure of developing this paper. [ABSTRACT FROM AUTHOR]

A graph G is called (r, s)-edge-choosable if for every assignment of sets of size r to the edges of G it is possible to choose for every edge an s-element subset from its set such that the subsets chosen for any pair of adjacent edges are disjoint. The list fractional chromatic index of a graph G is the infimum of all numbers r/s for which G is (r, s)-edge-choosable. Mohar [http://www.fmf.uni-lj.si/~mohar/(June 2003)] posed a problem asking whether every cubic graph is (7,2)-edge-choosable. In 2009, Cranston and West [SIAM J. Discrete Math., 23 (2009), pp. 872-881] showed that every 3-edge-colorable cubic graph is (7,2)-edge-choosable and gave a sufficient condition with the help of which they proved that many non-3-edge-colorable-cubic graphs are (7,2)-edge-choosable. In this paper we prove that every bridgeless cubic graph is (7,2)-edge-choosable. We show that this result cannot be improved in the family of all cubic graphs, in the sense that there exists a cubic graph with list fractional chromatic index 7/2. The original question of Mohar remains open, and we further pose several related problems. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with a set of vertices V and a set of edges E. Given an integer r, we assign at most r labels (representing "resources") to each vertex. We say that such an assignment is an r-configuration if, for each label c, the vertices labeled by c form a dominating set for G. In this paper, we are interested in the maximum number, Dr(G), of labels that can be assigned to the vertices of graph G by an r-configuration. The decision problem "D1(G) ≥ K?" (known as the domatic number problem) is NP-complete. We first investigate Dr(G) for general graphs and establish bounds on Dr(G) in terms of D1(G) and the minimum vertex degree of G. We then discuss Dr(G) for d-regular graphs. We clearly have Dr(G) ≤ r(d + 1). We show that the problem of testing if Dr(G) = r(d + 1) is solvable in polynomial time for d-regular graphs with ∣V∣ = 2(d + 1), but is NP-complete for those with ∣V∣ = a(d + 1) for some integer a ≥ 3. Finally, we discuss cubic (i.e., 3-regular) graphs. It is easy to show 2r ≤ Dr(G) ≤ 4r for cubic graphs. We show that the decision problem for D1(G) = K is co-NP-complete for K = 2 and is NP-complete for K = 4. Although there are many cubic graphs G with D1(G) = 2, surprisingly, every cubic graph has a 2-configuration with five labels, i.e., D2(G) ≥ 5 and such a 2-configuration can be constructed in polynomial time. We use this fact to show Dr(G) ≥. ⌊5r/2⌋ in general. [ABSTRACT FROM AUTHOR]

We establish that for any connected cubic graph G of order n > 16 the maximum P 3-matching in G consists of at least $$\frac{117}{152}n$$ vertices. [ABSTRACT FROM AUTHOR]