arXiv subjects

Peter Bradshaw

Publications and source records attributed to Peter Bradshaw.

At least 37 records · Page 2Linked to original sources

arXiv2024

Hamiltonicity of covering graphs of trees

In this paper, we consider covering graphs obtained by lifting a tree with a loop at each vertex as a voltage graph over a cyclic group. 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 $T$ whose edge labels are assigned uniformly at random from a finite set, 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.

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arXiv2024

List-avoiding orientations

Given a graph $G$ with a set $F(v)$ of forbidden values at each $v \in V(G)$, an $F$-avoiding orientation of $G$ is an orientation in which $deg^+(v) \not \in F(v)$ for each vertex $v$. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F(v)| < \frac{1}{2} deg(v)$ for each $v \in V(G)$, then $G$ has an $F$-avoiding orientation, and they showed that this statement is true when $\frac{1}{2}$ is replaced by $\frac{1}{4}$. In this paper, we take a step toward this conjecture by proving that if $|F(v)| < \lfloor \frac{1}{3} deg(v) \rfloor$ for each vertex $v$, then $G$ has an $F$-avoiding orientation. Furthermore, we show that if the maximum degree of $G$ is subexponential in terms of the minimum degree, then this coefficient of $\frac{1}{3}$ can be increased to $\sqrt{2} - 1 - o(1) \approx 0.414$. Our main tool is a new sufficient condition for the existence of an $F$-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi.

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arXiv2024

Hat guessing number and guaranteed subgraphs

The hat guessing number of a graph is a parameter related to the hat guessing game for graphs introduced by Winkler. In this paper, we show that graphs of sufficiently large hat guessing number must contain arbitrary trees and arbitrarily long cycles as subgraphs. More precisely, for each tree $T$, there exists a value $N = N(T)$ such that every graph that does not contain $T$ as a subgraph has hat guessing number at most $N$, and for each integer $c$, there exists a value $N' = N'(c)$ such that every graph with no cycle of length greater than $c$ has hat guessing number at most $N'$.

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arXiv2023

Fractional colorings of partial $t$-trees with no large clique

Dvořák and Kawarabayashi [European Journal of Combinatorics, 2017] asked, what is the largest chromatic number attainable by a graph of treewidth $t$ with no $K_r$ subgraph? In this paper, we consider the fractional version of this question. We prove that if $G$ has treewidth $t$ and clique number $2 \leq ω\leq t$, then $χ_f(G) \leq t + \frac{ω- 1}{t}$, and we show that this bound is tight for $ω= t$. We also show that for each value $0 < c < \frac{1}{2}$, there exists a graph $G$ of a large treewidth $t$ and clique number $ω= \lfloor (1 - c)t \rfloor$ satisfying $χ_f(G) \geq t + 1 + \frac{1}{2}\log(1-2c) + o(1)$, which is approximately equal to the upper bound for small values $c$.

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arXiv2023

Injective edge colorings of degenerate graphs and the oriented chromatic number

Given a graph $G$, an injective edge-coloring of $G$ is a function $ψ:E(G) \rightarrow \mathbb N$ such that if $ψ(e) = ψ(e')$, then no third edge joins an endpoint of $e$ and an endpoint of $e'$. The injective chromatic index of a graph $G$, written $χ_{inj}'(G)$, is the minimum number of colors needed for an injective edge coloring of $G$. In this paper, we investigate the injective chromatic index of certain classes of degenerate graphs. First, we show that if $G$ is a $d$-degenerate graph of maximum degree $Δ$, then $χ_{inj}'(G) = O(d^3 \log Δ)$. Next, we show that if $G$ is a graph of Euler genus $g$, then $χ_{inj}'(G) \leq (3+o(1))g$, which is tight when $G$ is a clique. Finally, we show that the oriented chromatic number of a graph is at most exponential in its injective chromatic index. Using this fact, we prove that the oriented chromatic number of a graph embedded on a surface of Euler genus $g$ has oriented chromatic number at most $O(g^{6400})$, improving the previously known upper bound of $2^{O(g^{\frac{1}{2} + ε})}$ and resolving a conjecture of Aravind and Subramanian.

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arXiv2023

Rainbow spanning trees in random subgraphs of dense regular graphs

We consider the following random model for edge-colored graphs. A graph $G$ on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting each edge of $G$ remain independently with probability $p$. Then, each edge of $G_p$ is colored uniformly at random from the set $[n-1]$. A result of Frieze and McKay (Random Structures and Algorithms, 1994) implies that when $G = K_n$ and $p = (2 + ε) \frac{\log n}{n}$ for some constant $ε> 0$, then $G_p$ almost surely contains a rainbow spanning tree. In this paper, we show that if $G$ is a $d$-regular $Ω(n)$-edge-connected graph, then when $p = (2 + ε) \frac {\log n}{d}$ for some constant $ε> 0$, $G_p$ almost surely contains a rainbow spanning tree. Our main tool is a new edge-replacement method for rainbow forests.

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arXiv2022

Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

We show that for any fixed integer $m \geq 1$, a graph of maximum degree $Δ$ has a coloring with $O(Δ^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which $O(Δ^{3/2})$ colors suffice, and acyclic colorings, for which $O(Δ^{4/3})$ colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph of maximum degree $Δ$ has a proper coloring with $O(Δ^{9/8})$ colors in which every bicolored subgraph is planar, as well as a proper coloring with $O(Δ^{13/12})$ colors in which every bicolored subgraph has treewidth at most $3$.

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arXiv2022

Separating the online and offline DP-chromatic numbers

The DP-coloring problem is a generalization of the list-coloring problem in which the goal is to find an independent transversal in a certain topological cover of a graph $G$. In the online DP-coloring problem, the cover of $G$ is revealed one component at a time, and the independent transversal of the cover must be constructed in parts based on incomplete information. Kim, Kostochka, Li, and Zhu asked whether the chromatic numbers corresponding to these two graph coloring problems can have an arbitrarily large difference in a single graph. We answer this question in the affirmative by constructing graphs for which the gap between the online DP-chromatic number and the offline DP-chromatic number is arbitrarily large.

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arXiv2022

Cooperative colorings of forests

Given a family $\mathcal G$ of graphs spanning a common vertex $V$, a cooperative coloring of $\mathcal G$ is a collection of one independent set from each graph of $\mathcal G$ such that the union of these independent sets equals $V$. We prove that when $d$ is large, there exists a family $\mathcal G$ of $(1+o(1)) \frac{\log d}{\log \log d}$ forests of maximum degree $d$ that admits no cooperative coloring, which significantly improves a result of Aharoni, Berger, Chudnovsky, Havet, and Jiang (Electronic Journal of Combinatorics, 2020). Our family $\mathcal G$ consists entirely of star forests, and we show that this value for $|\mathcal G|$ is asymptotically best possible in the case that $\mathcal G$ is a family of star forests.

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arXiv2021

Graph colorings with restricted bicolored subgraphs: II. The graph coloring game

We consider the graph coloring game, a game in which two players take turns properly coloring the vertices of a graph, with one player attempting to complete a proper coloring, and the other player attempting to prevent a proper coloring. We show that if a graph $G$ has a proper coloring in which the game coloring number of each bicolored subgraph is bounded, then the game chromatic number of $G$ is bounded. As a corollary to this result, we show that for two graphs $G_1$ and $G_2$ with bounded game coloring number, the Cartesian product $G_1 \square G_2$ has bounded game chromatic number, answering a question of X. Zhu. We also obtain an upper bound on the game chromatic number of the strong product $G_1 \boxtimes G_2$ of two graphs.

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arXiv2021

A rainbow connectivity threshold for random graph families

Given a family $\mathcal G$ of graphs on a common vertex set $X$, we say that $\mathcal G$ is rainbow connected if for every vertex pair $u,v \in X$, there exists a path from $u$ to $v$ that uses at most one edge from each graph in $\mathcal G$. We consider the case that $\mathcal G$ contains $s$ graphs, each sampled randomly from $G(n,p)$, with $n = |X|$ and $p = \frac{c \log n}{sn}$, where $c > 1$ is a constant. We show that when $s$ is sufficiently large, $\mathcal G$ is a.a.s. rainbow connected, and when $s$ is sufficiently small, $\mathcal G$ is a.a.s. not rainbow connected. We also calculate a threshold of $s$ for the rainbow connectivity of $\mathcal G$, and we show that this threshold is concentrated on at most three values, which are larger than the diameter of the union of $\mathcal G$ by about $\frac{\log n}{(\log \log n)^2}$. The same results also hold in a more traditional random rainbow setting, where we take a random graph $G\in G(n,p)$ with $p=\frac{c \log n}{n}$ ($c>1$) and color each edge of $G$ with a color chosen uniformly at random from the set $[s]$ of $s$ colors.

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arXiv2021

Robust Connectivity of Graphs on Surfaces

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. This problem is called maximum leaf number, and it is a well-known NP-hard problem. Throughout recent decades, this problem has received considerable attention, ranging from pure graph theoretic questions to practical problems related to the construction of wireless networks. Recently, a similar but stronger notion was defined by Bradshaw, Masařík, and Stacho [Flexible List Colorings in Graphs with Special Degeneracy Conditions, ISAAC 2020]. They introduced a new invariant for a graph $G$, called the robust connectivity and written $κ_ρ(G)$, defined as the minimum value $\frac{|R \cap Λ(T)|}{|R|}$ taken over all nonempty subsets $R\subseteq V(G)$, where $T = T(R)$ is a spanning tree on $G$ chosen to maximize $|R \cap Λ(T)|$. Large robust connectivity was originally used to show flexible choosability in non-regular graphs. In this paper, we investigate some interesting properties of robust connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound of $Ω(γ^{-\frac{1}{r}})$ 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 [A conjecture on planar graphs, 1979], which states that every planar graph on $n$ vertices contains an induced forest of size at least $n/2$.

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arXiv2021

From one to many rainbow Hamiltonian cycles

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 $(T,ϕ)$ such that $T \subseteq E(G)$ and $ϕ: T \rightarrow [n]$ is a bijection satisfying $e \in G_{ϕ(e)}$ for each $e \in T$. We call a transversal 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 $m \geq 300$, 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 $\frac{n}{2}$, then $\mathcal{G}$ has a Hamiltonian transversal: we show that, in this setting, $\mathcal{G}$ has exponentially many Hamiltonian transversals. Finally, we prove analogues of both of these theorems for transversals which form perfect matchings of $G$.

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arXiv2021

Transversals and bipancyclicity in bipartite graph families

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geq 4$, every balanced bipartite graph on $2n$ vertices in which each vertex in one color class has degree greater than $\frac{n}{2}$ and each vertex in the other color class has degree at least $\frac{n}{2}$ is bipancyclic. We prove a generalization of this theorem in the setting of graph transversals. Namely, we show that given a family $\mathcal{G}$ of $2n$ bipartite graphs on a common set $X$ of $2n$ vertices with a common balanced bipartition, if each graph of $\mathcal G$ has minimum degree greater than $\frac{n}{2}$ in one color class and minimum degree at least $\frac{n}{2}$ in the other color class, then there exists a cycle on $X$ of each even length $4 \leq \ell \leq 2n$ that uses at most one edge from each graph of $\mathcal G$. We also show that given a family $\mathcal G$ of $n$ bipartite graphs on a common set $X$ of $2n$ vertices meeting the same degree conditions, there exists a perfect matching on $X$ that uses exactly one edge from each graph of $\mathcal G$.

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arXiv2021

Flexible List Colorings in Graphs with Special Degeneracy Conditions

For a given $\varepsilon > 0$, we say that a graph $G$ is $\varepsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of color lists of size $k$ on $V(G)$, if a preferred color from a list is requested at any set $R$ of vertices, then at least $\varepsilon |R|$ of these requests are satisfied by some $L$-coloring. We consider the question of flexible choosability in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree $Δ$ that are $\varepsilon$-flexibly $Δ$-choosable for some $\varepsilon = \varepsilon(Δ) > 0$, which answers a question of Dvořák, Norin, and Postle [List coloring with requests, JGT 2019]. In particular, we show that for any $Δ\geq 3$, any graph of maximum degree $Δ$ that is not isomorphic to $K_{Δ+1}$ is $\frac{1}{6Δ}$-flexibly $Δ$-choosable. Our fraction of $\frac{1}{6 Δ}$ is within a constant factor of being the best possible. We also show that graphs of treewidth $2$ are $\frac{1}{3}$-flexibly $3$-choosable, answering a question of Choi et al.~[arXiv 2020], and we give conditions for list assignments by which graphs of treewidth $k$ are $\frac{1}{k+1}$-flexibly $(k+1)$-choosable. We show furthermore that graphs of treedepth $k$ are $\frac{1}{k}$-flexibly $k$-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 non-regular graphs of maximum degree $Δ$ are flexibly $(Δ- 1)$-degenerate.

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arXiv2020

An Incidence Result for Well-Spaced Atoms in all Dimensions

We prove an incidence result counting the $k$-rich $δ$-tubes induced by a well-spaced set of $δ$-atoms. Our result coincides with the bound that would be heuristically predicted by the Szemerédi--Trotter Theorem and holds in all dimensions $d \geq 2$.

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arXiv2020

Cops and robbers on directed and undirected abelian Cayley graphs

We show that the cop number of directed and undirected Cayley graphs on abelian groups has an upper bound of the form of $O(\sqrt{n})$, where $n$ is the number of vertices, by introducing a refined inductive method. With our method, we improve the previous upper bound on cop number for undirected Cayley graphs on abelian groups, and we establish an upper bound on the cop number of directed Cayley graphs on abelian groups. We also use Cayley graphs on abelian groups to construct new \emph{Meyniel extremal families}, which contain graphs of every order $n$ with cop number $Θ(\sqrt{n})$.

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arXiv2020

A note on the connected game coloring number

We consider the \emph{connected game coloring number} of a graph, introduced by Charpentier et al. as a game theoretic graph parameter that measures the degeneracy of a graph with respect to a certain two-player game played with an uncooperative adversary. We consider the connected game coloring number of graphs of bounded treedepth and of $k$-trees. In particular, we show that there exists an outerplanar $2$-tree with connected game coloring number of $5$, which answers a question from [C. Charpentier, H. Hocquard, E. Sopena, and X. Zhu. A connected version of the graph coloring game. \textit{Discrete Appl. Math.}, 2020].

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