Integer Flows and Circuit Covers of Graphs and Signed Graphs

The work in Chapter 2 is motivated by Tutte and Jaeger\u27s pioneering work on converting modulo flows into integer-valued flows for ordinary graphs. For a signed graphs (G, sigma), we first prove that for each k ∈ {lcub}2, 3{rcub}, if (G, sigma) is (k -- 1)-edge-connected and contains an even number of negative edges when k = 2, then every modulo k-flow of (G, sigma) can be converted into an integer-valued ( k + 1)-ow with a larger or the same support. We also prove that if (G, sigma) is odd-(2p+1)-edge-connected, then (G, sigma) admits a modulo circular (2 + 1/ p)-flows if and only if it admits an integer-valued circular (2 + 1/p)-flows, which improves all previous result by Xu and Zhang (DM2005), Schubert and Steffen (EJC2015), and Zhu (JCTB2015).;Shortest circuit cover conjecture is one of the major open problems in graph theory. It states that every bridgeless graph G contains a set of circuits F such that each edge is contained in at least one member of F and the length of F is at most 7/5∥E(G)∥. This concept was recently generalized to signed graphs by Macajova et al. (JGT2015). In Chapter 3, we improve their upper bound from 11∥E( G)∥ to 14/3 ∥E(G)∥, and if G is 2-edgeconnected and has even negativeness, then it can be further reduced to 11/3 ∥E(G)∥.;Tutte\u27s 3-flow conjecture has been studied by many graph theorists in the last several decades. As a new approach to this conjecture, DeVos and Thomassen considered the vectors as ow values and found that there is a close relation between vector S1-flows and integer 3-NZFs. Motivated by their observation, in Chapter 4, we prove that if a graph G admits a vector S1-flow with rank at most two, then G admits an integer 3-NZF.;The concept of even factors is highly related to the famous Four Color Theorem. We conclude this dissertation in Chapter 5 with an improvement of a recent result by Chen and Fan (JCTB2016) on the upperbound of even factors. We show that if a graph G contains an even factor, then it contains an even factor H with.;∥E(H)∥ ≥ 4/7 (∥ E(G)∥+1)+ 1/7 ∥V2 (G)∥, where V2( G) is the set of vertices of degree two

Flows on Simplicial Complexes

Given a graph $G$, the number of nowhere-zero \ZZ_q-flows $\phi_G(q)$ is known to be a polynomial in $q$. We extend the definition of nowhere-zero \ZZ_q-flows to simplicial complexes $\Delta$ of dimension greater than one, and prove the polynomiality of the corresponding function $\phi_{\Delta}(q)$ for certain $q$ and certain subclasses of simplicial complexes.Comment: 10 pages, to appear in Discrete Mathematics and Theoretical Computer Science (proceedings of FPSAC'12

An extension of König's theorem to graphs with no odd-K4

Mathematics

French Engineers and the Machinery of Society: X-Crise and the Development of the first Macroeconomic Models in the Nineteen Thirties

The purpose of this article is to highlight the originality of X-Crise, an Ecole Polytechnique association formed in nineteen thirties' France. Firstly il analyses the factors leading the French engineers to look collectively, for the first time of their history, at the big politicoeconomic problems of their epoch. Two factors seem particularly relevant on this matter : their epistemology and value system on one hand, and their perception of the theoretical and political impotence at that time on the other hand.Secondly this paper presents the main original theoretical elements that arose from the group's deliberations. Among them, it exposes the Guillaume brothers' macroeconomic model [1932], one of the very first macroeconomic model produced in France, and Potron's first application to economics of Perron-Frobenius's theorem (Potron [1911] and [1935]).French Engineers - Macroeconomic models - Pragmatism - Mechanics - Perron-Frobenius's theorem

Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume "Recent Trends in Combinatorics