Nowhere-zero 3-flows in products of graphs

This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte's 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte's 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. Extending Tutte's flows conjectures, Jaeger's 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's 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's Circular Flow Conjecture and Tutte's 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's 3-Flow Conjecture. In Chapter 4 and Chapter 5, utilizing properties of additive bases and contractible configurations, we reduced the Esperet et al's 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's Circular Flow Conjecture with k=2 holds asymptotically almost surely for random 9-regular graphs.

Nowhere-zero 3-flows in Cayley graphs of order pq2

Tutte’s 3-Flow Conjecture suggests that every bridgeless graph with no 3-edge-cut can have its edges directed and labelled by the numbers 1 or 2 in such a way that at each vertex the sum of incoming values equals the sum of outgoing values. In this paper we show that Tutte’s 3-Flow Conjecture is true for Cayley graphs of groups whose Sylow 2-subgroup is a direct factor of the group; in particular, it is true for Cayley graphs of nilpotent groups. This improves a recent result of Potočnik et al. (Discrete Math. 297:119–127, 2005) concerning nowhere-zero 3-flows in abelian Cayley graphs.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

: There are four prominent product graphs in graph theory: Cartesian, strong, direct and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted G o H, for some digraphs G and H. The paper concludes by proving a cancellation property for the lexicographic product of digraphs G, H, A, and B: if G o H = A o B and /V(G)/ = /V(A)/, then G = A. It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments.

Flow number and circular flow number of signed cubic graphs

Considering a set of intervals on the real line, an interval graph records these intervals as nodes and their intersections as edges. Identifying (i.e. merging) pairs of nodes in an interval graph results in a multiple-interval graph. Given only the nodes and the edges of the multiple-interval graph without knowing the underlying intervals, we are interested in the following questions. Can one determine how many intervals correspond to each node? Can one compute a walk over the multiple-interval graph nodes that reflects the ordering of the original intervals? These questions are closely related to linked-read DNA sequencing, where barcodes are assigned to long molecules whose intersection graph forms an interval graph. Each barcode may correspond to multiple molecules, which complicates downstream analysis, and corresponds to the identification of nodes of the corresponding interval graph. Resolving the above graph-theoretic problems would facilitate analyses of linked-reads sequencing data, through enabling the conceptual separation of barcodes into molecules and providing, through the molecules order, a skeleton for accurately assembling the genome. Here, we propose a framework that takes as input an arbitrary intersection graph (such as an overlap graph of barcodes) and constructs a heuristic approximation of the ordering of the original intervals.

This work presents the employment of graph theory for cryptographic applications. A Cartesian product graph (CPG) is used to propose new symmetric encryption schemes. On proposed encryption schemes, the ciphertexts of the plaintexts are sent to the receiver entity as the CPGs. The CPG is created based on two graphs G and H that are generated depending on the random choices of the plaintext bits. These bits are used as vertices of graphs G and H. The purpose of using the CPG is to increase the security level of the symmetric encryption schemes in comparison with other encryption schemes. Two study cases of the proposed symmetric Cartesian product graphic encryption (SCPGE) schemes are presented as new experimental results. The security issues of the proposed SCPGE schemes are determined. The SCPGE schemes consider as new insights for more secure communications.

Graph products for configuration processing of space structures

Group distance magic and antimagic graphs

We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products. We show that the independent sets induced by the base graph are the only maximum-size independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size. Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4Eaiaaic % dacaGGSaGaeSOjGSKaaiilaiaadkhacqGHsislcaaIXaGaaiyFamaa % CaaaleqabaGaamOBaaaakiaacYcaaaa!3FA3! $$\{ 0, \ldots ,r - 1\} ^n ,$$ generalizing some useful results from the % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4Eaiaaic % dacaGGSaGaaGymaiaac2hadaahaaWcbeqaaiaad6gaaaaaaa!3B33! $$\{ 0,1\} ^n $$ case.

About some robustness and complexity properties of [formula omitted]-graphs networks

The tensor product is a fundamental mathematical concept with applications spanning linear algebra, graph theory, quantum computing, and representation theory. In graph theory, the tensor product provides a framework for analyzing structural relationships, particularly through the study of complete graphs, which yield complex networks from simple structures. Closely related is the Kronecker product of matrices, an essential tool for investigating tensor products via adjacency matrices. The Kronecker product preserves key algebraic properties, including linearity, distributivity, and associativity, and has played a central role in matrix analysis, systems theory, and signal processing. This work presents the definitions and core properties of the tensor product, supported by illustrative examples with complete graphs, and explores the Kronecker product along with its fundamental properties and theorems. By combining theoretical foundations with applications, the study offers both conceptual insights and practical perspectives on these algebraic constructions.

Cycle cover property and CPP=SCC property are not equivalent

We show that if the well-known 5-Flow Conjecture of Tutte is not true, then the problem to determine whether a (cubic) graph admits a nowhere-zero 5-flow is NP-complete. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 1–11, 1998