Edita Rollová

Homomorphisms of Signed Graphs

A signed graph [G,Σ] is a graph G together with an assignment of signs + and - to all the edges of G where Σ is the set of negative edges. Furthermore [G,Σ1] and [G,Σ2] are considered to be equivalent if the symmetric difference of Σ1 and Σ2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph homomorphisms where he showed how some well-known conjectures can be captured using this notion. A signed graph [G,Σ] is said to map to [H,Σ1] if there is an equivalent signed graph [G,Σ] of [G,Σ] and a mapping i¾?:VGi¾?VH such that i if xy∈EG then i¾?xi¾?y∈EH and iixy∈Σ if and only if i¾?xi¾?y∈Σ1. The chromatic number of a signed graph [G,Σ] can then be defined as the smallest order of a homomorphic image of [G,Σ]. Capturing the notion of graph homomorphism order, signed graph homomorphisms provide room for extensions and strengthenings of most homomorphism and coloring theories on graphs. Thus this paper is the first general study of signed graph homomorphisms. In this work, our focus would be on the relation of homomorphisms of signed graphs with minors of signed graphs. After a thorough introduction to the concept we show that the notion of signed graph homomorphism on the set of signed graphs whose underlying graph is bipartite already captures the standard notion of graph homomorphism. We prove that the largest planar signed clique is of order 8. For the maximum chromatic number of planar signed graphs we give the lower bound of 10 and the upper bound of 48. We determine this maximum for some other families such as outerplanar signed graphs. Finally, reformulating Hadwigers conjecture in the language of homomorphism of signed graphs whose underlying graph is bipartite, we show that while some stronger form of the conjecture holds for small chromatic number, such strengthening of the conjecture would not hold for large chromatic numbers. This could be regarded as a first indication that perhaps Hadwigers conjecture only holds for small chromatic numbers.

Homomorphisms of planar signed graphs to signed projective cubes

We conjecture that every signed graph of unbalanced girth 2g, whose underlying graph is bipartite and planar, admits a homomorphism to the signed projective cube of dimension 2g1. Our main result is to show that for a given g, this conjecture is equivalent to the corresponding case (k = 2g) of a conjecture of Seymour claiming that every planar k-regular multigraph with no odd edge-cut of less than k edges is k-edge-colorable. To this end, we exhibit several properties of signed projective cubes and establish a folding lemma for planar even signed graphs.

Circuit Covers of Signed Graphs

We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with circuits. As in the case of graphs, a signed graph has a signed circuit cover only when it admits a nowhere-zero integer flow. In the present article, we establish the existence of a universal coefficient q∈R such that every signed graph G that admits a nowhere-zero integer flow has a signed circuit cover of total length at most qi¾?|EG|. We show that if G is bridgeless, then qi¾?9, and in the general case qi¾?11.

Nowhere-Zero Flows on Signed Complete and Complete Bipartite Graphs

Bouchets conjecture asserts that each signed graph which admits a nowhere-zero flow has a nowhere-zero 6-flow. We verify this conjecture for two basic classes of signed graphs-signed complete and signed complete bipartite graphs by proving that each such flow-admissible graph admits a nowhere-zero 4-flow and we characterise those which have a nowhere-zero 2-flow and a nowhere-zero 3-flow.

Nowhere-Zero Flows in Signed Series-Parallel Graphs

Bouchet conjectured in 1983 that each signed graph that admits a nowhere-zero flow has a nowhere-zero 6-flow. We prove that the conjecture is true for all signed series-parallel graphs. Unlike the unsigned case, the restriction to series-parallel graphs is nontrivial; in fact, the result is tight for infinitely many graphs.

Nowhere-zero flows in Cartesian bundles of graphs

We extend the results of Imrich and Skrekovski [J. Graph Theory 43 (2003) 93-98] concerning nowhere-zero flows in Cartesian product graphs to twisted Cartesian products, that is, Cartesian bundles. Our main result states that every Cartesian bundle of two graphs without isolated vertices has a nowhere-zero 4-flow.

On the flow numbers of signed complete and complete bipartite graphs

Abstract We determine the flow numbers of signed complete and signed complete bipartite graphs.

Perfect Matchings of Regular Bipartite Graphs

Let G be a regular bipartite graph and X⊆E(G). We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph (G,X), that is a graph G with exactly the edges from X being negative, is not equivalent to (G,∅). In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2-cycle-cover such that each cycle contains an odd number of negative edges.

A new proof of Seymour's 6-flow theorem

Tuttes famous 5-flow conjecture asserts that every bridgeless graph has a nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero 6-flow. Here we give (two versions of) a new proof of Seymours Theorem. Both are roughly equal to Seymours in terms of complexity, but they offer an alternative perspective which we hope will be of value.