Nowhere-zero flows on signed regular graphs

We study the flow spectrum ${\cal S}(G)$ and the integer flow spectrum $\overline{{\cal S}}(G)$ of signed $(2t+1)$-regular graphs. We show that if $r \in {\cal S}(G)$, then $r = 2+\frac{1}{t}$ or $r \geq 2 + \frac{2}{2t-1}$. Furthermore, $2 + \frac{1}{t} \in {\cal S}(G)$ if and only if $G$ has a $t$-factor. If $G$ has a 1-factor, then $3 \in \overline{{\cal S}}(G)$, and for every $t \geq 2$, there is a signed $(2t+1)$-regular graph $(H,\sigma)$ with $3 \in \overline{{\cal S}}(H)$ and $H$ does not have a 1-factor. If $G$ $(\not = K_2^3)$ is a cubic graph which has a 1-factor, then $\{3,4\} \subseteq {\cal S}(G) \cap \overline{{\cal S}}(G)$. Furthermore, the following four statements are equivalent: (1) $G$ has a 1-factor. (2) $3 \in {\cal S}(G)$. (3) $3 \in \overline{{\cal S}}(G)$. (4) $4 \in \overline{{\cal S}}(G)$. There are cubic graphs whose integer flow spectrum does not contain 5 or 6, and we construct an infinite family of bridgeless cubic graphs with integer flow spectrum $\{3,4,6\}$. We show that there are signed graphs where the difference between the integer flow number and the flow number is greater than or equal to 1, disproving a conjecture of Raspaud and Zhu. The paper concludes with a proof of Bouchet's 6-flow conjecture for Kotzig-graphs.Comment: 24 pages, 4 figures; final version; to appear in European J. Combinatoric

Flows on Signed Graphs

This dissertation focuses on integer flow problems within specific signed graphs. The theory of integer flows, which serves as a dual problem to vertex coloring of planar graphs, was initially introduced by Tutte as a tool related to the Four-Color Theorem. This theory has been extended to signed graphs. In 1983, Bouchet proposed a conjecture asserting that every flow-admissible signed graph admits a nowhere-zero 6-flow. To narrow dawn the focus, we investigate cubic signed graphs in Chapter 2. We prove that every flow-admissible 3-edge-colorable cubic signed graph admits a nowhere-zero 10-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero 10-flow. As a byproduct of this research, we also demonstrate that every flow-admissible hamiltonian signed graph can admit a nowhere-zero 8-flow. In Chapter 3, we delve into triangularly connected signed graphs. Here, A triangle-path in a graph G is defined as a sequence of distinct triangles $T_1,T_2,\ldots,T_m$ in G such that for any i, j with $1\leq i \u3c j \leq m$, $|E(T_i)\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j \u3e i+1$. We categorize a connected graph $G$ as triangularly connected if it can be demonstrated that for any two nonparallel edges $e$ and $e\u27$, there exists a triangle-path $T_1T_2\cdots T_m$ such that $e\in E(T_1)$ and $e\u27\in E(T_m)$. For ordinary graphs, Fan {\it et al.} characterized all triangularly connected graphs that admit nowhere-zero $3$-flows or $4$-flows. Corollaries of this result extended to integer flow in certain families of ordinary graphs, such as locally connected graphs due to Lai and certain types of products of graphs due to Imrich et al. In this dissertation, we extend Fan\u27s result for triangularly connected graphs to signed graphs. We proved that a flow-admissible triangularly connected signed graph $(G,\sigma)$ admits a nowhere-zero $4$-flow if and only if $(G,\sigma)$ is not the wheel $W_5$ associated with a specific signature. Moreover, this result is proven to be sharp since we identify infinitely many unbalanced triangularly connected signed graphs that can admit a nowhere-zero 4-flow but not 3-flow.\\ Chapter 4 investigates integer flow problems within $K_4$-minor free signed graphs. A minor of a graph $G$ refers to any graph that can be derived from $G$ through a series of vertex and edge deletions and edge contractions. A graph is considered $K_4$-minor free if $K_4$ is not a minor of $G$. While Bouchet\u27s conjecture is known to be tight for some signed graphs with a flow number of 6. Kompi\v{s}ov\\u27{a} and M\\u27{a}\v{c}ajov\\u27{a} extended those signed graph with a specific signature to a family \M, and they also put forward a conjecture that suggests if a flow-admissible signed graph does not admit a nowhere-zero 5-flow, then it belongs to \M. In this dissertation, we delve into the members in \M that are $K_4$-minor free, designating this subfamily as $\N$. We provide a proof demonstrating that every flow-admissible, $K_4$-minor free signed graph admits a nowhere-zero 5-flow if and only if it does not belong to the specified family $\N$

Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications

A Not-All-Equal (NAE) decomposition of a graph $G$ is a decomposition of the vertices of $G$ into two parts such that each vertex in $G$ has at least one neighbor in each part. Also, a 1-in-Degree decomposition of a graph $G$ is a decomposition of the vertices of $G$ into two parts $A$ and $B$ such that each vertex in the graph $G$ has exactly one neighbor in part $A$. Among our results, we show that for a given graph $G$, if $G$ does not have any cycle of length congruent to 2 mod 4, then there is a polynomial time algorithm to decide whether $G$ has a 1-in-Degree decomposition. In sharp contrast, we prove that for every $r$, $r\geq 3$, for a given $r$-regular bipartite graph $G$ determining whether $G$ has a 1-in-Degree decomposition is $\mathbf{NP}$-complete. These complexity results have been especially useful in proving $\mathbf{NP}$-completeness of various graph related problems for restricted classes of graphs. In consequence of these results we show that for a given bipartite 3-regular graph $G$ determining whether there is a vector in the null-space of the 0,1-adjacency matrix of $G$ such that its entries belong to $\{\pm 1,\pm 2\}$ is $\mathbf{NP}$-complete. Among other results, we introduce a new version of {Planar 1-in-3 SAT} and we prove that this version is also $\mathbf{NP}$-complete. In consequence of this result, we show that for a given planar $(3,4)$-semiregular graph $G$ determining whether there is a vector in the null-space of the 0,1-incidence matrix of $G$ such that its entries belong to $\{\pm 1,\pm 2\}$ is $\mathbf{NP}$-complete.Comment: To appear in Algorithmic

A unified approach to construct snarks with circular flow number 5

The well-known 5-flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5-flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5 are known. In the first part of this paper, we summarise some of these methods and we propose new ones based on variations of the known constructions. Afterwards, we prove that all such methods are nothing but particular instances of a more general construction that we introduce into detail. In the second part, we consider many instances of this general method and we determine when our method permits to obtain a snark with circular flow number 5. Finally, by a computer search, we determine all snarks having circular flow number 5 up to 36 vertices. It turns out that all such snarks of order at most 34 can be obtained by using our method, and that the same holds for 96 of the 98 snarks of order 36 with circular flow number 5.Comment: 27 pages; submitted for publicatio

Weighted Modulo Orientations of Graphs

This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte\u27s 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte\u27s 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. Extending Tutte\u27s flows conjectures, Jaeger\u27s Circular Flow Conjecture (1981) says every 4k-edge-connected graph admits a modulo (2k+1)-orientation, that is, an orientation such that the indegree is congruent to outdegree modulo (2k+1) at every vertex. Note that the k=1 case of Circular Flow Conjecture coincides with the 3-Flow Conjecture, and the case of k=2 implies the 5-Flow Conjecture. This work is devoted to providing some partial results on these problems. In Chapter 2, we study the problem of modulo 5-orientation for given multigraphic degree sequences. We prove that a multigraphic degree sequence d=(d1,..., dn) has a realization G with a modulo 5-orientation if and only if di≤1,3 for each i. In addition, we show that every multigraphic sequence d=(d1,..., dn) with min{1≤i≤n}di≥9 has a 9-edge-connected realization that admits a modulo 5-orientation for every possible boundary function. Jaeger conjectured that every 9-edge-connected multigraph admits a modulo 5-orientation, whose truth would imply Tutte\u27s 5-Flow Conjecture. Consequently, this supports the conjecture of Jaeger. In Chapter 3, we show that there are essentially finite many exceptions for graphs with bounded matching numbers not admitting any modulo (2k+1)-orientations for any positive integers t. We additionally characterize all infinite many graphs with bounded matching numbers but without a nowhere-zero 3-flow. This partially supports Jaeger\u27s Circular Flow Conjecture and Tutte\u27s 3-Flow Conjecture. In 2018, Esperet, De Verclos, Le and Thomass introduced the problem of weighted modulo orientations of graphs and indicated that this problem is closely related to modulo orientations of graphs, including Tutte\u27s 3-Flow Conjecture. In Chapter 4 and Chapter 5, utilizing properties of additive bases and contractible configurations, we reduced the Esperet et al\u27s edge-connectivity lower bound for some (signed) graphs families including planar graphs, complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families. In Chapter 6, we show that the assertion of Jaeger\u27s Circular Flow Conjecture with k=2 holds asymptotically almost surely for random 9-regular graphs

The Parity Hamiltonian Cycle Problem

Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that the walk may use an edge more than once. First, we give a complete characterization of the graphs which have PHCs, and give a linear time algorithm to find a PHC, in which every edge appears at most four times, in fact. In contrast, we show that finding a PHC is NP-hard if a closed walk is allowed to use each edge at most z times for each z=1,2,3 (PHCz for short), even when a given graph is two-edge connected. We then further investigate the PHC3 problem, and show that the problem is in P when an input graph is four-edge connected. Finally, we are concerned with three (or two)-edge connected graphs, and show that the PHC3 is in P for any C_>=5-free or P6-free graphs. Note that the Hamiltonian cycle problem is known to be NP-hard for those graph classes.Comment: 29 pages, 16 figure

The arithmetic of polynomial dynamical pairs

We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical Andr\'e-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.Comment: 246 pages, 12 pictures, 5 table

Discrete Geometry

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Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor