Related papers: The Local Structure Theorem for Graph Minors with …
The subdivision graph $S(\Sigma)$ of a connected graph $\Sigma$ is constructed by adding a vertex in the middle of each edge. In a previous paper written with Cheryl E. Praeger, we characterised the graphs $\Sigma$ such that $S(\Sigma)$ is…
Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes…
Counting small patterns in a large dataset is a fundamental algorithmic task. The most common version of this task is subgraph/homomorphism counting, wherein we count the number of occurrences of a small pattern graph $H$ in an input graph…
We pursue the study of edge-irregulators of graphs, which were recently introduced in [Fioravantes et al. Parametrised Distance to Local Irregularity. IPEC, 2024]. That is, we are interested in the parameter Ie(G), which, for a given graph…
Lov\'{a}sz and Cherkassky discovered independently that, if $G$ is a finite graph and $T\subseteq V(G)$ such that the degree $d_G(v)$ is even for every vertex $v\in V(G)\setminus T$, then the maximum number of edge-disjoint paths which are…
We show an $\Omega\big(\Delta^{\frac{1}{3}-\frac{\eta}{3}}\big)$ lower bound on the runtime of any deterministic distributed $\mathcal{O}\big(\Delta^{1+\eta}\big)$-graph coloring algorithm in a weak variant of the \LOCAL\ model. In…
A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…
Given a graph family $\mathbb{H}$, let ${\rm SPEX}(n,\mathbb{H}_{\rm sub})$ denote the set of $n$-vertex $\mathbb{H}$-subdivision-free graphs with the maximum spectral radius. In this paper, we investigate the problem of graph subdivision…
Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A…
A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary…
We prove a coarse version of Halin's Grid Theorem: Every one-ended, locally finite graph that contains the disjoint union of infinitely many rays as an asymptotic minor also contains the half-grid as an asymptotic minor. More generally, we…
A proof labelling scheme for a graph class $\mathcal{C}$ is an assignment of certificates to the vertices of any graph in the class $\mathcal{C}$, such that upon reading its certificate and the certificates of its neighbors, every vertex…
The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We…
The $2$-cell embeddings of graphs on closed surfaces have been widely studied. It is well known that ($2$-cell) embedding a given graph $G$ on a closed orientable surface is equivalent to cyclically ordering the edges incident to each…
A dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G) \setminus D$ is adjacent to at least one vertex in $D$. A set $L\subseteq V(G)$ is a locating set of $G$ if every vertex in $V(G) \setminus L$ has…
A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of…
The stability number of a graph $G$, denoted as $\alpha(G)$, is the maximum size of an independent (stable) set in $G$. Semidefinite programming (SDP) methods, which originated from Lov\'asz's theta number and expanded through…
The Graph Minor Theorem of Robertson and Seymour implies a finite set of obstructions for any minor closed graph property. We show that there are only three obstructions to knotless embedding of size 23, which is far fewer than the 92 of…
For every $r \in \mathbb{N}$, let $\theta_r$ denote the graph with two vertices and $r$ parallel edges. The $\theta_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $\theta_r$. This…
A multigraph in which adjacent vertices have different degrees is called locally irregular. The locally irregular edge coloring is an edge coloring of a multigraph $G$ in which every color induces a locally irregular submultigraph of $G$.…