34 research outputs found
Characterization of Line-Consistent Signed Graphs
The line graph of a graph with signed edges carries vertex signs. A
vertex-signed graph is consistent if every circle (cycle, circuit) has positive
vertex-sign product. Acharya, Acharya, and Sinha recently characterized
line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with
the naturally induced vertex signature, are consistent. Their proof applies
Hoede's relatively difficult characterization of consistent vertex-signed
graphs. We give a simple proof that does not depend on Hoede's theorem as well
as a structural description of line-consistent signed graphs.Comment: 5 pages. V2 defines sign of a walk and corrects statement of Theorem
4 ("is balanced and" was missing); also minor copyeditin
Coloring permutation-gain graphs
Correspondence colorings of graphs were introduced in 2018 by Dvořák and Postle as a generalization of list colorings of graphs which generalizes ordinary graph coloring. Kim and Ozeki observed that correspondence colorings generalize various notions of signed-graph colorings which again generalizes ordinary graph colorings. In this note we state how correspondence colorings generalize Zaslavsky's notion of gain-graph colorings and then formulate a new coloring theory of permutation-gain graphs that sits between gain-graph coloring and correspondence colorings. Like Zaslavsky's gain-graph coloring, our new notion of coloring permutation-gain graphs has well defined chromatic polynomials and lifts to colorings of the regular covering graph of a permutation-gain graph
On Colorings and Orientations of Signed Graphs
A classical theorem independently due to Gallai and Roy states that a graph G has a proper k-coloring if and only if G has an orientation without coherent paths of length k. An analogue of this result for signed graphs is proved in this article
Graphs Without a 2C3-Minor and Bicircular Matroids Without a U3,6-Minor
In this note we characterize all graphs without a 2C3-minor. A consequence of this result is a characterization of the bicircular matroids with no U3,6-minor
Characterization of a Family of Rotationally Symmetric Spherical Quadrangulations
A spherical quadrangulation is an embedding of a graph G in the sphere in which each facial boundary walk has length four. Vertices that are not of degree four in G are called curvature vertices. In this paper we classify all spherical quadrangulations with n-fold rotational symmetry (n ≥ 3) that have minimum degree 3 and the least possible number of curvature vertices, and describe all such spherical quadrangulations in terms of nets of quadrilaterals. The description reveals that such rotationally symmetric quadrangulations necessarily also have a pole-exchanging symmetry
The Family of Bicircular Matroids Closed Under Duality
We characterize the 3-connected members of the intersection of the class of bicircular and cobi- circular matroids. Aside from some exceptional matroids with rank and corank at most 5, this class consists of just the free swirls and their minors
Describing Quasi-Graphic Matroids
The class of quasi-graphic matroids recently introduced by Geelen, Gerards,
and Whittle generalises each of the classes of frame matroids and
lifted-graphic matroids introduced earlier by Zaslavsky. For each biased graph
Zaslavsky defined a unique lift matroid
and a unique frame matroid , each on ground set . We
show that in general there may be many quasi-graphic matroids on and
describe them all. We provide cryptomorphic descriptions in terms of subgraphs
corresponding to circuits, cocircuits, independent sets, and bases. Equipped
with these descriptions, we prove some results about quasi-graphic matroids. In
particular, we provide alternate proofs that do not require 3-connectivity of
two results of Geelen, Gerards, and Whittle for 3-connected matroids from their
introductory paper: namely, that every quasi-graphic matroid linearly
representable over a field is either lifted-graphic or frame, and that if a
matroid has a framework with a loop that is not a loop of then is
either lifted-graphic or frame. We also provide sufficient conditions for a
quasi-graphic matroid to have a unique framework.
Zaslavsky has asked for those matroids whose independent sets are contained
in the collection of independent sets of while containing
those of , for some biased graph . Adding a
natural (and necessary) non-degeneracy condition defines a class of matroids,
which we call biased graphic. We show that the class of biased graphic matroids
almost coincides with the class of quasi-graphic matroids: every quasi-graphic
matroid is biased graphic, and if is a biased graphic matroid that is not
quasi-graphic then is a 2-sum of a frame matroid with one or more
lifted-graphic matroids
Hamilton cycles in bidirected complete graphs
Zaslavsky observed that the topics of directed cycles in directed graphs and alternating cycles in edge 2-colored graphs have a common generalization in the study of coherent cycles in bidirected graphs. There are classical theorems by Camion, Harary and Moser, Häggkvist and Manoussakis, and Saad which relate strong connectivity and Hamiltonicity in directed "complete" graphs and edge 2-colored "complete" graphs. We prove two analogues to these theorems for bidirected "complete" signed graphs
Odd solutions to systems of inequalities coming from regular chain groups
Hoffman’s theorem on feasible circulations and Ghouila-Houry’s theorem on feasible tensions are classical results of graph theory. Camion generalized these results to systems of inequalities over regular chain groups. An analogue of Camion’s result is proved in which solutions can be forced to be odd valued. The obtained result also generalizes the results of Pretzel and Youngs as well as Slilaty. It is also shown how Ghouila-Houry’s result can be used to give a new proof of the graph- coloring theorem of Minty and Vitaver