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Daniel W. Cranston

Publications and source records attributed to Daniel W. Cranston.

At least 19 recordsLinked to original sources
arXiv2026

A Linear Kernel for Independent Set Reconfiguration in Planar Graphs

Fix a positive integer $r$, and a graph $G$ that is $K_{3,r}$-minor-free. Let $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a ``token'' on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated ``token jumping'', removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ into $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kamiński, and Ono (2014) to be fixed-parameter tractable. That is, the problem can be solved in time $f(k)\cdot Poly(n)$, for some function $f$ and polynomial $Poly(n)$, where $n$ denotes the order of $G$. Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a $K_{3,r}$-minor-free graph (for some fixed positive integer $r$) into an equivalent problem on a ($K_{3,r}$-minor-free) graph with order $O(k)$. This answers positively a question of Bousquet, Mouawad, Nishimura, and Siebertz (2024) and improves the recent quadratic kernel of Cranston, Mühlenthaler, and Peyrille (2026). For planar graphs, we further strengthen this upper bound to get a kernel of size at most $39k$.

math.CO↗
arXiv2026

Orientations of $10$-Edge-Connected Planar Multigraphs and Applications

A graph is called strongly $\Z_{2k+1}$-connected if for each boundary function $β: V(G)\mapsto \Z_{2k+1}$ with $\sum_{v\in V(G)}β(v)\equiv 0\pmod{2k+1}$, there exists an orientation $D$ of $G$ such that $d_D^+(v) - d_D^-(v) \equiv β(v) \pmod{2k+1}$ for each $v \in V(G)$. We show that every planar multigraph with $5$ edge-disjoint spanning trees is strongly $\Z_{5}$-connected. This verifies a special case of the Additive Base Conjecture when restricted to planar graphs. Hence, every $10$-edge-connected directed planar graph admits an antisymmetric $\Z_5$-flow. So, by duality, every orientation of a planar graph of girth at least $10$ admits a homomorphism to a $5$-vertex tournament. Our result also gives a new proof of the known result that every planar graph of girth at least $10$ has a homomorphism to the $5$-cycle.

math.CO↗
arXiv2026

Reconfiguration of Nowhere-zero Flows

Fix an abelian group $A$, a graph $G$, and nowhere-zero $A$-flows $f'$ and $f''$ on $G$. Now $f'$ and $f''$ are \emph{$A$-flow-adjacent} if there exists a cycle $C$ in $G$ such that $f'(e)-f''(e)=0$ for all edges $e\notin E(C)$. And $f'$ and $f''$ are \emph{$A$-flow-equivalent} if there exists a sequence $f_0,\ldots,f_s$ of $A$-flows such that $f_0=f'$, $f_s=f''$, and $f_i$ and $f_{i-1}$ are $A$-flow-adjacent for all $i\in[s]$. Given a group $A$, we seek conditions on a graph $G$ such that all $A$-flows on $G$ are pairwise $A$-flow-equivalent; in this case, we say that $G$ is \emph{$A$-flow-connected}. Analogously, we define $k$-flow-connectedness for nowhere-zero (integer) $k$-flows. The notions of $A$-flow-connectedness and $k$-flow-connectedness were first investigated by Esperet et al., who showed, among other results, that every $2$-edge-connected graph is $A$-flow-connected whenever $A=\mathbb{Z}_2^8$ or $|A| \ge 1.15\times 10^{694}$. In this paper, we first characterize the graphs that are $\mathbb{Z}_3$-flow-connected and that are $3$-flow-connected. We show that every 2-edge-connected graph is $A$-flow-connected if and only if this is true for every 2-edge-connected cubic graphs. We show that all cubic bipartite graphs are $\mathbb{Z}_4$-flow-connected, and construct other cubic graphs that are and are not $\mathbb{Z}_4$-flow-connected. We conjecture that every Eulerian graph is $k$-flow-connected and $A$-flow-connected whenever $k$ or $|A|$ is even; and provide evidence for this conjecture. Finally, we consider $4$-edge-connected graphs $G$. Here, we show that $G$ is $A$-flow-connected whenever $|A|\ge 5.3\times 10^6$.

math.CO↗
arXiv2026

Disjoint Correspondence Colorings for $K_5$-Minor-free Graphs

Thomassen famously proved that every planar graph is 5-choosable. We explore variants of this result, focusing on finding disjoint correspondence colorings, in the more general class of $K_5$-minor-free graphs. Correspondence colorings generalize list colorings as follows. Given a graph $G$ and a positive integer $t$, a correspondence $t$-cover $\textbf{M}$ assigns to each $v\in V(G)$ a set of allowable colors $\{1_v,\ldots,t_v\}$ and to each edge $vw\in E(G)$ a matching between $\{1_v,\ldots,t_v\}$ and $\{1_w,\ldots,t_w\}$. An $\textbf{M}$-coloring $φ$ picks for each vertex $v$ a color $φ(v)$ (from the set $\{1_v,\ldots,t_v\}$) such that for each edge $vw\in E(G)$ the colors $φ(v),φ(w)$ are not matched to each other. Two $\textbf{M}$-colorings $φ_1,φ_2$ of $G$ are called disjoint if $φ_1(v)\neφ_2(v)$ for all $v\in V(G)$. For every $K_5$-minor-free graph $G$ and every correspondence 6-cover $\textbf{M}$ of $G$, we construct 3 pairwise disjoint $\textbf{M}$-colorings $φ_1,φ_2,φ_3$. In contrast, we provide examples of $K_5$-minor-free graphs and correspondence 5-covers $\textbf{M}$ that do not admit 3 disjoint $\textbf{M}$-colorings.

math.CO↗
arXiv2025

Progress on Albertson's Conjecture

Albertson conjectured that every graph with chromatic number $r$ has crossing number at least the crossing number of the complete graph $K_r$. This conjecture was proved for $r\le 12$ by Albertson, Cranston, and Fox; for $r\le 16$ by Barát and Tóth; and for $r\le 18$ by Ackerman. Here we verify it for $r\le 24$; we also greatly restrict the possibilities for counterexamples when $r\in\{25,26\}$. In addition, we strengthen earlier work bounding the order of a minimum counterexample for each choice of $r$: we exclude the possibility that $|G|\ge 2.82r$ and exclude the possibility that $1.228r\le |G|\le 1.768r$. Finally, as $r$ grows, we extend the lower end of this range of excluded orders for a minimum counterexample. In particular: if $r\ge 125{,}000$, then we exclude the possibility that $1.10r\le |G|\le 1.768r$; and if $r\ge 825{,}000$, then we exclude the possibility that $1.05r\le |G|\le 1.768r$.

math.CO↗
arXiv2025

Equitably Coloring Planar and Outerplanar Graphs

A proper $s$-coloring of an $n$-vertex graph is \emph{equitable} if every color class has size $\lfloor{n/s}\rfloor$ or $\lceil{n/s}\rceil$. A necessary condition to have an equitable $s$-coloring is that every vertex $v$ appears in an independent set of size at least $\lfloor{n/s}\rfloor$. That is $\min_{v\in V(G)}α_v\ge \lfloor{n/s}\rfloor$. Various authors showed that when $G$ is a tree and $s\ge 3$ this obvious necessary condition is also sufficient. Kierstead, Kostochka, and Xiang asked whether this result holds more generally for all outerplanar graphs. We show that the answer is No when $s=3$, but that the answer is Yes when $s\ge 6$. The case $s\in\{4,5\}$ remains open. We also prove an analogous result for planar graphs, with a necessary and sufficient hypothesis. Fix $s\ge 40$. Let $G$ be a planar graph, and let $w_0,w_1$ be its $2$ vertices with largest degrees. If there exist disjoint independent sets $I_0, I_1$ such that $|I_0|=\lfloor{n/s}\rfloor$ and $|I_1| = \lfloor{(n+1)/s}\rfloor$ and $w_0,w_1\in I_0\cup I_1$, then $G$ has an equitable $s$-coloring.

math.CO↗
arXiv2025

10-list Recoloring of Planar Graphs

Fix a planar graph $G$ and a list-assignment $L$ with $|L(v)|=10$ for all $v\in V(G)$. Let $α$ and $β$ be $L$-colorings of $G$. A recoloring sequence from $α$ to $β$ is a sequence of $L$-colorings, beginning with $α$ and ending with $β$, such that each successive pair in the sequence differs in the color on a single vertex of $G$. We show that there exists a constant $C$ such that for all choices of $α$ and $β$ there exists a recoloring sequence $σ$ from $α$ to $β$ that recolors each vertex at most $C$ times. In particular, $σ$ has length at most $C|V(G)|$. This confirms a conjecture of Dvořák and Feghali. For our proof, we introduce a new technique for quickly showing that many configurations are reducible. We believe this method may be of independent interest and will have application to other problems in this area.

math.CO↗
arXiv2025

Reconfiguration of List Colourings

Given a proper (list) colouring of a graph $G$, a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours, hence maintaining a proper colouring. Suppose that each vertex $v$ has its own private list $L(v)$ of allowed colours such that $|L(v)|\ge \mbox{deg}(v)+1$. We prove that if $G$ is connected and its maximum degree $Δ$ is at least $3$, then for any two proper $L$-colourings in which at least one vertex can be recoloured, one can be transformed to the other by a sequence of $O(|V(G)|^2)$ recolouring steps. We also show that reducing the list-size of a single vertex $w$ to $\mbox{deg}(w)$ can lead to situations where the space of proper $L$-colourings is `shattered'. Our results can be interpreted as showing a sharp phase transition in the Glauber dynamics of proper $L$-colourings of graphs. This constitutes a `local' strengthening and generalisation of a result of Feghali, Johnson, and Paulusma, which considered the situation where the lists are all identical to $\{1,\ldots,Δ+1\}$.

math.CO↗
arXiv2024

Sharp Bounds on Lengths of Linear Recolouring Sequences

A recolouring sequence, between $k$-colourings $α$ and $β$ of a graph $G$, transforms $α$ into $β$ by recolouring one vertex at a time, such that after each recolouring step we again have a proper $k$-colouring of $G$. The diameter of the $k$-recolouring graph, $\textrm{diam}~\mathcal{C}_k(G)$, is the maximum over all pairs $α$ and $β$ of the minimum length of a recolouring sequence from $α$ to $β$. Much previous work has focused on determining the asymptotics of $\textrm{diam}~\mathcal{C}_k(G)$: Is it $Θ(|G|)$? Is it $Θ(|G|^2)$? Or even larger? Here we focus on graphs for which $\textrm{diam}~\mathcal{C}_k(G)=Θ(|G|)$, and seek to determine more precisely the multiplicative constant implicit in the $Θ()$. In particular, for each $k\ge 3$, for all positive integers $p$ and $q$ we exactly determine $\textrm{diam}~\mathcal{C}_k(K_{p,q})$, up to a small additive constant. We also sharpen a recolouring lemma that has been used in multiple papers, proving an optimal version. This improves the multiplicative constant in various prior results. Finally, we investigate plausible relationships between similar reconfiguration graphs.

math.CO↗
arXiv2024

List Packing and Correspondence Packing of Planar Graphs

For a graph $G$ and a list assignment $L$ with $|L(v)|=k$ for all $v$, an $L$-packing consists of $L$-colorings $φ_1,\cdots,φ_k$ such that $φ_i(v)\neφ_j(v)$ for all $v$ and all distinct $i,j\in\{1,\ldots,k\}$. Let $χ^{\star}_{\ell}(G)$ denote the smallest $k$ such that $G$ has an $L$-packing for every $L$ with $|L(v)|=k$ for all $v$. Let $\mathcal{P}_k$ denote the set of all planar graphs with girth at least $k$. We show that (i) $χ^{\star}_{\ell}(G)\le 8$ for all $G\in \mathcal{P}_3$ and (ii) $χ^{\star}_{\ell}(G)\le 5$ for all $G\in \mathcal{P}_4$ and (iii) $χ^{\star}_{\ell}(G)\le 4$ for all $G\in \mathcal{P}_5$. Part (i) makes progress on a problem of Cambie, Cames van Batenburg, Davies, and Kang. We also construct outerplanar graphs $G$ such that $χ^{\star}_{\ell}(G)=4$, which matches the known upper bound $χ^{\star}_{\ell}(G)\le 4$ for all outerplanar graphs. Finally, we consider the analogue of $χ^{\star}_{\ell}$ for correspondence coloring, $χ^{\star}_c$. In fact, all bounds stated above for $χ^{\star}_{\ell}$ also hold for $χ^{\star}_c$.

math.CO↗
arXiv2024

Token Jumping in Planar Graphs has Linear Sized Kernels

Let $G$ be a planar graph and $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a "token" on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated "token jumping", removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ to $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kamiński, and Ono to be fixed-parameter tractable. That is, when $k$ is fixed, the problem can be solved in time polynomial in the order of $G$. Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a planar graph into an equivalent problem on a (planar) graph with order $O(k)$.

cs.DM↗
arXiv2024

A simple quadratic kernel for Token Jumping on surfaces

The problem \textsc{Token Jumping} asks whether, given a graph $G$ and two independent sets of \emph{tokens} $I$ and $J$ of $G$, we can transform $I$ into $J$ by changing the position of a single token in each step and having an independent set of tokens throughout. We show that there is a polynomial-time algorithm that, given an instance of \textsc{Token Jumping}, computes an equivalent instance of size $O(g^2 + gk + k^2)$, where $g$ is the genus of the input graph and $k$ is the size of the independent sets.

cs.DS↗
arXiv2024

Kempe Classes and Almost Bipartite Graphs

Let $G$ be a graph and $k$ be a positive integer, and let $Kc(G, k)$ denote the number of Kempe equivalence classes for the $k$-colorings of $G$. In 2006, Mohar noted that $Kc(G, k) = 1$ if $G$ is bipartite. As a generalization, we show that $Kc(G, k) = 1$ if $G$ is formed from a bipartite graph by adding any number of edges less than $\binom{\lceil k/2\rceil}2+\binom{\lfloor k/2\rfloor}2$. We show that our result is tight (up to lower order terms) by constructing, for each $k \geq 8$, a graph $G$ formed from a bipartite graph by adding $(k^2+8k-45+1)/4$ edges such that $Kc(G, k) \geq 2$. This refutes a recent conjecture of Higashitani--Matsumoto.

math.CO↗
arXiv2024

Cliques in Squares of Graphs with Maximum Average Degree less than 4

Hocquard, Kim, and Pierron constructed, for every even integer $D\ge 2$, a 2-degenerate graph $G_D$ with maximum degree $D$ such that $ω(G_D^2)=\frac52D$. We prove for (a) all 2-degenerate graphs $G$ and (b) all graphs $G$ with $\mbox{mad}(G)<4$, upper bounds on the clique number $ω(G^2)$ of $G^2$ that match the lower bound given by this construction, up to small additive constants. We show that if $G$ is 2-degenerate with maximum degree $D$, then $ω(G^2)\le \frac52D+72$ (with $ω(G^2)\le \frac52D+60$ when $D$ is sufficiently large). And if $G$ has $\mbox{mad}(G)<4$ and maximum degree $D$, then $ω(G^2)\le \frac52D+532$. Thus, the construction of Hocquard et al. is essentially best possible. Our proofs introduce a "token passing" technique to derive crucial information about non-adjacencies in $G$ of vertices that are adjacent in $G^2$. This is a powerful technique for working with such graphs that has not previously appeared in the literature.

math.CO↗
arXiv2024

Bounding Clique Size in Squares of Planar Graphs

Wegner conjectured that if $G$ is a planar graph with maximum degree $Δ\ge 8$, then $χ(G^2)\le \left\lfloor \frac32Δ\right\rfloor +1$. This problem has received much attention, but remains open for all $Δ\ge 8$. Here we prove an analogous bound on $ω(G^2)$: If $G$ is a plane graph with $Δ(G)\ge 36$, then $ω(G^2)\le \lfloor\frac32Δ(G)\rfloor+1$. In fact, this is a corollary of the following lemma, which is our main result. If $G$ is a plane graph with $Δ(G)\ge 19$ and $S$ is a maximal clique in $G^2$ with $|S|\ge Δ(G)+20$, then there exist $x,y,z\in V(G)$ such that $S=\{w:|N[w]\cap\{x,y,z\}|\ge 2\}$.

math.CO↗
arXiv2024

Proper Conflict-free Coloring of Graphs with Large Maximum Degree

A proper coloring of a graph is \emph{conflict-free} if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petruševski, and Škrekovski proved that every graph $G$ has a proper conflict-free coloring with at most $5Δ(G)/2$ colors and conjectured that $Δ(G)+1$ colors suffice for every connected graph $G$ with $Δ(G)\ge 3$. Our first main result is that even for list-coloring, $\left\lceil 1.6550826Δ(G)+\sqrt{Δ(G)}\right\rceil$ colors suffice for every graph $G$ with $Δ(G)\ge 10^{8}$; we also prove slightly weaker bounds for all graphs with $Δ(G)\ge 750$. These results follow from our more general framework on proper conflict-free list-coloring of a pair consisting of a graph $G$ and a "conflict" hypergraph ${\mathcal H}$. As another corollary of our results in this general framework, every graph has a proper $(\sqrt{30}+o(1))Δ(G)^{1.5}$-list-coloring such that every bi-chromatic component is a path on at most three vertices, where the number of colors is optimal up to a constant factor. Our proof uses a fairly new type of recursive counting argument called Rosenfeld counting, which is a variant of the Lovász Local Lemma or entropy compression. We also prove an asymptotically optimal result for a fractional analogue of our general framework for proper conflict-free coloring for pairs of a graph and a conflict hypergraph. A corollary states that every graph $G$ has a fractional $(1+o(1))Δ(G)$-coloring such that every fractionally bi-chromatic component has at most two vertices. In particular, it implies that the fractional analogue of the conjecture of Caro et al.\ holds asymptotically in a strong sense.

math.CO↗
arXiv2024

Planar Graphs with Homomorphisms to the 9-cycle

We study the problem of finding homomorphisms into odd cycles from planar graphs with high odd-girth. The Jaeger-Zhang conjecture states that every planar graph of odd-girth at least $4k+1$ admits a homomorphism to the odd cycle $C_{2k+1}$. The $k=1$ case is the well-known Grötzsch's $3$-coloring theorem. For general $k$, in 2013 Lovász, Thomassen, Wu, and Zhang showed that it suffices to have odd-girth at least $6k+1$. Improvements are known for $C_5$ and $C_7$ in [Combinatorica 2017, SIDMA 2020, Combinatorica 2022]. For $C_9$ we improve this hypothesis by showing that it suffices to have odd-girth 23. Our main tool is a variation on the potential method applied to modular orientations. This allows more flexibility when seeking reducible configurations. The same techniques also prove some results on circular coloring of signed planar graphs.

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