arXiv subjects

Amena Assem

Publications and source records attributed to Amena Assem.

5 recordsLinked to original sources

arXiv2026

Algebraic Characterizations for Minors of Finite Graphs via Flow Transformation Monoid Division and Embedding

We prove three theorems on the flow monoids of finite graphs. First, we show that a non-empty finite graph G = (V, E) is connected if and only if its flow monoid contains a constant map on V, equivalently, if and only if it contains all constant maps on V. Second, we give a new characterization of graph minors in terms of division of flow transformation monoids, together with an algebraic crossing condition that detects edges between the vertex sets being contracted. Third, we strengthen this to an embedded-copy theorem: a graph M is a minor of G if and only if, subject to analogous crossing conditions, the flow transformation monoid of M is realized as the induced action of a subsemigroup of the ambient flow monoid of G, this subsemigroup being a monoid with a local idempotent identity.

cs.DM↗

arXiv2025

Edge-disjoint linkage in infinite graphs

In 1980, Thomassen stated his weak linkage conjecture: for an odd positive integer k, if a graph G is k-edge-connected, then, for any collection of k pairs of vertices {s_1,t_1}, ..., {s_k,t_k} in G, not necessarily distinct, there are pairwise edge-disjoint paths P_1,...,P_k in G, with P_i joining s_i and t_i. In 1991, Huck proved that the conclusion holds if G is finite and (k+1)-edge-connected. We prove that Huck's theorem holds also for all infinite graphs, extending and improving a result of Ok, Richter and Thomassen for 1-ended, locally finite graphs. A novel key tool in the proof is the Linking Fan Proposition proved in Section 3. To show the potential and usefulness of this proposition in other contexts, we apply it in the last section to prove a new result, similar to a result of Thomassen, on the existence of 2k-edge-connected finite immersions in (2k+1)-edge-connected infinite graphs. We then use this to prove that an edge-connectivity of 2k+1 is sufficient for infinite graphs to admit a k-arc-connected orientation. This is only within 1 of the longstanding conjecture of Nash-Williams from 1960 that an edge-connectivity of 2k should be enough.

math.CO↗

arXiv2023

The Nash-Williams orientation theorem for graphs with countably many ends

Nash-Williams proved in 1960 that a finite graph admits a $k$-arc-connected orientation if and only if it is $2k$-edge-connected, and conjectured that the same result should hold for all infinite graphs, too. Progress on Nash-Williams's problem was made by C. Thomassen, who proved in 2016 that all $8k$-edge-connected infinite graphs admit a $k$-arc connected orientation, and by the first author, who recently showed that edge-connectivity of $4k$ suffices for locally-finite, 1-ended graphs. In the present article, we establish the optimal bound $2k$ in Nash-Williams's conjecture for all locally finite graphs with countably many ends.

math.CO↗

arXiv2023

Towards Nash-Williams Orientation Conjecture for Infinite Graphs

In 1960 Nash-Williams proved that an edge-connectivity of 2k is sufficient for a finite graph to have a k-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that 8k-edge-connected infinite graphs admit a $k$-arc connected orientation. Here we improve this result for the class of $1$-ended locally-finite graphs and show that an edge-connectivity of 4k is enough in that case. Crucial to this improvement are results presented in a separate paper, by the same author of this paper, on the key concept of the lifting graph, extending results by Ok, Richter, and Thomassen.

math.CO↗

arXiv2022

Analysis of the Lifting Graph

The `lifting` or `splitting-off` operation on graphs is performed by deleting two edges sv and sw having a common end s and adding a new edge between v and w. Such a lift is considered good if it preserves a certain local edge-connectivity between the pairs of vertices different from the vertex s at which lifting takes place. The operation is important for inductive proofs concerning edge-connectivity, and can be seen widely applied in the literature on connectivity augmentation, network design, orientation (of finite and infinite graphs), and edge-disjoint linkage. It was studied by Lovasz, who used the term splitting-off, and Mader, who used the term lifting. They proved the first two significant results on it, in 1976 and 1978 respectively, showing the existence of a good lift under certain conditions. Then it was used and studied by other researchers, through the 1980s, both for undirected and directed graphs. In particular, it was investigated further by Frank who proved in 1992 that there are floor of deg(s)/2 disjoint good lifts. Motivated by the applications, a new method for studying the operation was introduced by Jordan in the late 1990s. He defined and studied the structure of the `non-admissibility` graph, which is the complement of the lifting graph; the subject of this paper. He proved a number of significant structural results on it, which he applied to connectivity augmentation. Independently, in 2016, Thomassen defined the `lifting graph`, and called its complement the `bad graph`, to apply it in finding orientations of infinite graphs. Later in the same year, Thomassen with Ok and Richter extended the study, and applied their results to linkages in infinite graphs. Here we give a more comprehensive analysis of the structure of the lifting graph.

math.CO↗