Ladislav Stacho’s research while affiliated with Simon Fraser University and other places

Alon and Krivelevich conjectured that if G is a bipartite graph of maximum degree $\Delta$, then the choosability (or list chromatic number) of G satisfies $\chi_{\ell}(G) = O \left ( \log \Delta \right )$. Currently, the best known upper bound for $\chi_{\ell}(G)$ is $(1 + o(1)) \frac{\Delta}{\log \Delta}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\varepsilon = 10^{-3}$, every bipartite graph G of sufficiently large maximum degree $\Delta$ satisfies $\chi_{\ell}(G) < (\frac{4}{5} -\varepsilon) \frac{\Delta}{\log \Delta}$. This improved upper bound suggests that list coloring is fundamentally different for bipartite graphs than for triangle-free graphs and hence gives a step toward solving the conjecture of Alon and Krivelevich.

This paper considers upper bounds on the oriented chromatic number, $\chi_o$, of graphs in terms of their maximum degree $\Delta$ and/or their degeneracy d. In particular we show that asymptotically, $\chi_o \leq \chi_2 f(d) 2^d$ where $f(d) \geq (\frac{1}{\log_2(e) -1} + \epsilon) d^2$ and $\chi_2 \leq 2^{\frac{f(d)}{d}}$. This improves a result of MacGillivray, Raspaud, and Swartz of the form $\chi_o \leq 2^{\chi_2} -1$. The rest of the paper is devoted to improving prior bounds for $\chi_o$ in terms of $\Delta$ and d by refining the asymptotic arguments involved.

Given a graph G and a family $\mathcal {G} = \{G_1,\ldots ,G_n\}$ of subgraphs of G, a transversal of $\mathcal {G}$ is a pair $(T,\phi )$ such that $T \subseteq E(G)$ and $\phi : T \rightarrow [n]$ is a bijection satisfying $e \in G_{\phi (e)}$ for each $e \in T$. We call a transversal $(T, \phi )$ Hamiltonian if T corresponds to the edge set of a Hamiltonian cycle in G. We show that, under certain conditions on the maximum degree of G and the minimum degrees of the $G_i \in \mathcal {G}$, for every $\mathcal {G}$ which contains a Hamiltonian transversal, the number of Hamiltonian transversals contained in $\mathcal {G}$ is bounded below by a function of G’s maximum degree. This generalizes a theorem of Thomassen stating that, for $m \ge 300$, no m-regular graph is uniquely Hamiltonian. We also extend Joos and Kim’s recent result that, if $G = K_{n}$ and each $G_i \in \mathcal {G}$ has minimum degree at least n/2, then $\mathcal {G}$ has a Hamiltonian transversal: we show that, in this setting, $\mathcal {G}$ has factorially many Hamiltonian transversals. Finally, we prove analogues of both of these theorems for transversals which form perfect matchings in G.

In this paper, we consider covering graphs obtained by lifting trees as voltage graphs over cyclic groups. We generalize a tool of Hell, Nishiyama, and Stacho, known as the billiard strategy, for constructing Hamiltonian cycles in the covering graphs of paths. We show that our extended tool can be used to provide new sufficient conditions for the Hamiltonicity of covering graphs of trees that are similar to those of Batagelj and Pisanski and of Hell, Nishiyama, and Stacho. Next, we focus specifically on covering graphs obtained from trees lifted as voltage graphs over cyclic groups $\mathbb Z_p$ of large prime order p. We prove that for a given reflexive tree T with random nonzero voltage labels from $\mathbb Z_p$ on its edges, the corresponding lift is almost surely Hamiltonian for a large enough prime-ordered cyclic group $\mathbb Z_p$. Finally, we show that if a reflexive tree T is lifted over a group $\mathbb Z_p$ of a large prime order, then for any assignment of nonzero elements of $\mathbb Z_p$ to the edges of T, the corresponding cover of T has a large circumference.

For a given ε> 0 $\varepsilon \gt 0$, we say that a graph G G is ε $\varepsilon$‐flexibly k k‐choosable if the following holds: for any assignment L L of color lists of size k k on V( G) V(G), if a preferred color from a list is requested at any set R R of vertices, then at least ε∣ R∣ $\varepsilon | R|$ of these requests are satisfied by some L L‐coloring. We consider the question of flexible choosability in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree Δ ${\rm{\Delta }}$ that are ε $\varepsilon$‐flexibly Δ ${\rm{\Delta }}$‐choosable for some ε= ε( Δ)> 0 $\varepsilon =\varepsilon ({\rm{\Delta }})\gt 0$, which answers a question of Dvořák Norin, and Postle [List coloring with requests, JGT 2019]. In particular, we show that for any Δ≥ 3 ${\rm{\Delta }}\ge 3$, any graph of maximum degree Δ ${\rm{\Delta }}$ that is not isomorphic to KΔ+ 1 ${K}_{{\rm{\Delta }}+1}$ is 16 Δ $\frac{1}{6{\rm{\Delta }}}$‐flexibly Δ ${\rm{\Delta }}$‐choosable. Our fraction of 16 Δ $\frac{1}{6{\rm{\Delta }}}$ is within a constant factor of being the best possible. We also show that graphs of treewidth 2 are 1 3 $\frac{1}{3}$‐flexibly 3‐choosable, answering a question of Choi et al. [Flexibility of planar graphs‐sharpening the tools to get lists of size four, DAM 2022], and we give conditions for list assignments by which graphs of treewidth k k are 1k+1 $\frac{1}{k+1}$‐flexibly (k+1) (k+1)‐choosable. We show furthermore that graphs of treedepth k k are 1 k $\frac{1}{k}$‐flexibly k k‐choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well‐understood class of exceptions, 3‐connected nonregular graphs of maximum degree Δ ${\rm{\Delta }}$ are flexibly (Δ−1) $({\rm{\Delta }}-1)$‐degenerate.

We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth g g and minimum degree δ $\delta$ is at least 1 g ( δ − 1 ) ⌊ g − 1 4 ⌋ $\frac{1}{g}{(\delta -1)}^{\lfloor \frac{g-1}{4}\rfloor }$. We establish similar results for directed graphs. While exposing several reasons for conjecturing that the exponent 1 4 g $\frac{1}{4}g$ in this lower bound cannot be improved to ( 1 4 + ε ) g $(\frac{1}{4}+\varepsilon )g$, we are also able to prove that it cannot be increased beyond 3 8 g $\frac{3}{8}g$. This is established by considering a certain family of Ramanujan graphs. In our proof of this bound, we also show that the “weak” Meyniel's conjecture holds for expander graph families of bounded degree.

Let Λ(T) denote the set of leaves in a tree T. One natural problem is to look for a spanning tree T of a given graph G such that Λ(T) is as large as possible. Recently, a similar but stronger notion called the robust connectivity of a graph G was introduced, which is defined as the minimum value |R∩Λ(T)||R| taken over all nonempty subsets R⊆V(G), where T=T(R) is a spanning tree on G chosen to maximize |R∩Λ(T)|. We prove a tight asymptotic bound of Ω(γ-1r) for the robust connectivity of r-connected graphs of Euler genus γ. Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman.