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Register a new userWhich Cubic Graphs have Quadrangulated Spherical Immersions?
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We consider spherical quadrangulations, i.e., graph embeddings in the sphere, in which every face has boundary walk of length 4, and all vertices have degree 3 or 4. Interpreting each degree 4 vertex as a crossing, these embeddings can also be thought of as transversal immersions of cubic graphs which we refer to as the extracted graphs. We also consider quadrangulations of the disk in which interior vertices have degree 3 or 4 and boundary vertices have degree 2 or 3. First, we classify all such quadrangulations of the disk. Then, we provide four methods for constructing spherical quadrangulations, two of which use quadrangulations of the disk as input. Two of these methods provide one-parameter families of quadrangulations, for which we prove that the sequence of isomorphism types of extracted graphs is periodic. We close with a description of computer computations which yielded spherical quadrangulations for all but three cubic multigraphs on eight vertices.
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For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free graphs? We are interested in identifying graphs $F$ with the property that, roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of $n^{3/2}$ edges, then the density of $F$ in $G$, after a suitable normalization, is approximately at least the density of $F$ in an $epsilon$-regular approximation of $G$. In recent work, motivated by applications in extremal and additive combinatorics, we showed that $C_5$ has this property. Here we construct a family of graphs with the property.
Following a problem posed by Lovasz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that is, for groups $G=la a,b| a^2=1, b^s=1, (ab)^3=1, etc. ra$ generated by an involution $a$ and an element $b$ of order $sgeq3$ such that their product $ab$ has order 3. More precisely, it is shown that the Cayley graph $X=Cay(G,{a,b,b^{-1}})$ has a Hamilton cycle when $|G|$ (and thus $s$) is congruent to 2 modulo 4, and has a long cycle missing only two vertices (and thus necessarily a Hamilton path) when $|G|$ is congruent to 0 modulo 4.
A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in general if it is true for apex graphs and doublecross graphs. In another paper, two of us solved the apex case, but the doublecross case remained open. Here we solve the doublecross case; that is, we prove that every two-edge-connected doublecross cubic graph is three-edge-colourable. The proof method is a variant on the proof of the four-colour theorem.
In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In addition, in the appendix by K. Sackel the F(q)-rational points of its graded augmentation variety are shown to coincide with (q+1)-colorings of the dual graph.
An (r,alpha)-bounded excess flow ((r,alpha)-flow) in an orientation of a graph G=(V,E) is an assignment of a real flow value between 1 and r-1 to every edge. Rather than 0 as in an actual flow, some flow excess, which does not exceed alpha may accumulate in any vertex. Bounded excess flows suggest a generalization of Circular nowhere zero flows, which can be regarded as (r,0)-flows. We define (r,alpha) as Stronger or equivalent to (s,beta) If the existence of an (r,alpha)-flow in a cubic graph always implies the existence of an (s,beta)-flow in the same graph. Then we study the structure of the two-dimensional flow strength poset. A major role is played by the Trace parameter: tr(r,alpha)=(r-2alpha) divided by (1-alpha). Among points with the same trace the stronger is the one with the larger r (an r-cnzf is of trace r). About one half of the article is devoted to proving the main result: Every cubic graph admits a (3.5,0.5)-flow. tr(3.5,0.5)=5 so it can be considered a step in the direction of the 5-flow Conjecture. Our result is best possible for all cubic graphs while the seemingly stronger 5-flow Conjecture only applies to bridgeless graphs. We strongly rely on the notion of k-weak bisections, defined and studied in: L. Esperet, G. Mazzuoccolo and M. Tarsi Flows and Bisections in Cubic Graphs J. Graph Theory, 86(2) (2017), 149-158.
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