arXiv subjects

Douglas B. West

Publications and source records attributed to Douglas B. West.

At least 37 records · Page 2Linked to original sources

arXiv2019

Largest 2-regular subgraphs in 3-regular graphs

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally, every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the extremal multigraphs.

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arXiv2018

The Slow-coloring Game on Sparse Graphs: $k$-Degenerate, Planar, and Outerplanar

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $\frac73n$ when $G$ is outerplanar.

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arXiv2018

On-line size Ramsey number for monotone k-uniform ordered paths with uniform looseness

An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The $t$-color on-line size Ramsey number $\tilde R_t (G)$ of an ordered hypergraph $G$ is the minimum number of edges Builder needs to play (on a large ordered set of vertices) to force Painter using $t$ colors to produce a monochromatic copy of $G$. The monotone tight path $P_r^{(k)}$ is the ordered hypergraph with $r$ vertices whose edges are all sets of $k$ consecutive vertices. We obtain good bounds on $\tilde R_t (P_r^{(k)})$. Letting $m=r-k+1$ (the number of edges in $P_r^{(k)}$), we prove $m^{t-1}/(3\sqrt t)\le\tilde R_t (P_r^{(2)})\le tm^{t+1}$. For general $k$, a trivial upper bound is ${R \choose k}$, where $R$ is the least number of vertices in a $k$-uniform (ordered) hypergraph whose $t$-colorings all contain $P_r^{(k)}$ (and is a tower of height $k-2$). We prove $R/(k\lg R)\le\tilde R_t(P_r^{(k)})\le R(\lg R)^{2+ε}$, where $ε$ is any positive constant and $t(m-1)$ is sufficiently large. Our upper bounds improve prior results when $t$ grows faster than $m/\log m$. We also generalize our results to $\ell$-loose monotone paths, where each successive edge begins $\ell$ vertices after the previous edge.

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arXiv2018

Cut-edges and regular factors in regular graphs of odd degree

We study $2k$-factors in $(2r+1)$-regular graphs. Hanson, Loten, and Toft proved that every $(2r+1)$-regular graph with at most $2r$ cut-edges has a $2$-factor. We generalize their result by proving for $k\le(2r+1)/3$ that every $(2r+1)$-regular graph with at most $2r-3(k-1)$ cut-edges has a $2k$-factor. Both the restriction on $k$ and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly $2r-3(k-1)+1$ cut-edges but no $2k$-factor. For $k>(2r+1)/3$, there are graphs without cut-edges that have no $2k$-factor, as studied by Bollobás, Saito, and Wormald.

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arXiv2018

Ramsey Numbers of Interval 2-chromatic Ordered Graphs

An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the vertex set. The $t$-color Ramsey number $R_t(G)$ of an ordered graph $G$ is the minimum number of vertices of an ordered complete graph such that every edge-coloring from a set of $t$ colors contains a monochromatic copy of $G$ such that the copy of $G$ preserves the original ordering on $G$. An ordered graph is $k$-ichromatic if it has interval chromatic number $k$. We obtain lower bounds linear in the number of vertices for the Ramsey numbers of certain classes of 2-ichromatic ordered graphs. We also determine the exact value of the $t$-color Ramsey number for two families of 2-ichromatic ordered graphs, and we prove a linear upper bound for a class of 2-ichromatic ordered graphs.

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arXiv2018

Extremal problems on saturation for the family of $k$-edge-connected graphs

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number and extremal number of $\mathcal{F}$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in\mathbb{N}$, let $\mathcal{F}_k$ and $\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n,\mathcal{F}_k)=(k-1)n-{k\choose2}$, we prove $sat(n,\mathcal{F}'_k)=(k-1)(n-1)-\lfloor{\frac {n}{k+1}}\rfloor{k-1 \choose 2}$. We also prove $ex(n,\mathcal{F}'_k)=(k-1)n-{k\choose2}$ and characterize when equality holds. Finally, we give a lower bound on the spectral radius for $\mathcal{F}_k$-saturated and $\mathcal{F}'_k$-saturated graphs.

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arXiv2017

The Unit Acquisition Number of a Graph

Let $G$ be a graph with nonnegative integer weights. A {\it unit acquisition move} transfers one unit of weight from a vertex to a neighbor that has at least as much weight. The {\it unit acquisition number} of a graph $G$, denoted $a_u(G)$, is the minimum size that the set of vertices with positive weight can be reduced to via successive unit acquisition moves when starting from the configuration in which every vertex has weight $1$. For a graph $G$ with $n$ vertices and minimum degree $k$, we prove $a_u(G)\le (n-1)/k$, with equality for complete graphs and $C_5$. Also $a_u(G)$ is at most the minimum size of a maximal matching in $G$, with equality on an infinite family of graphs. Furthermore, $a_u(G)$ is bounded by the maximum degree and by $\sqrt{n-1}$ when $G$ is an $n$-vertex tree with diameter at most $4$. We also construct arbitrarily large trees with maximum degree $5$ having unit acquisition number $1$, obtain a linear-time algorithm to compute the acquisition number of a caterpillar, and show that graphs with diameter $2$ have unit acquisition number $1$ except for $C_5$ and the Petersen graph.

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arXiv2017

Online Sum-Paintability: Slow-Coloring of Trees

The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the remaining vertices, scoring $|M|$ points. Painter then gives a color to a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. Painter's goal is to minimize the total score; Lister seeks to maximize it. The score that each player can guarantee doing no worse than is the sum-color cost of $G$, written $\mathring{\rm s}(G)$. We develop a linear-time algorithm to compute $\mathring{\rm s}(G)$ when $G$ is a tree, enabling us to characterize the $n$-vertex trees with the largest and smallest values. Our algorithm also computes on trees the interactive sum choice number, a parameter recently introduced by Bonamy and Meeks.

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arXiv2017

Online Paintability: The Slow-Coloring Game

The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the uncolored vertices, scoring $|M|$ points. Painter then gives a color to a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. Painter and Lister want to minimize and maximize the total score, respectively. The best score that each player can guarantee is the sum-color cost of $G$, written $\mathring{\mathrm{s}}(G)$. The game is an online variant of online sum list coloring. We proe $\frac{|V(G)|}{2α(G)} + \frac{1}{2} \leq \frac{\mathring{\mathrm{s}}(G)}{|V(G)|} \leq \max\left\{ \frac{|V(H)|}{α(H)} : H \subset G\right\}$, where $α(G)$ is the independence number, and we study when equality holds in the bounds. We compute $\mathring{\mathrm{s}}(G)$ for graphs with $α(G) = 2$. Among $n$-vertex graphs, we prove that $\mathring{\mathrm{s}}$ is minimized by the star and maximized by the path. We also obtain good bounds on $\mathring{\mathrm{s}}(K_{r,s})$.

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arXiv2016

An Introduction to the Discharging Method via Graph Coloring

We provide a "how-to" guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph coloring as examples. Along the way, we present some new proofs and new problems.

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arXiv2016

Fractional and Circular Separation Dimension of Graphs

The separation dimension of a graph $G$, written $π(G)$, is the minimum number of linear orderings of $V(G)$ such that every two nonincident edges are "separated" in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the fractional separation dimension $π_f(G)$, which is the minimum of $a/b$ such that some $a$ linear orderings (repetition allowed) separate every two nonincident edges at least $b$ times. In contrast to separation dimension, fractional separation dimension is bounded: always $π_f(G)\le 3$, with equality if and only if $G$ contains $K_4$. There is no stronger bound even for bipartite graphs, since $π_f(K_{m,m})=π_f(K_{m+1,m})=\frac{3m}{m+1}$. We also compute $π_f(G)$ for cycles and some complete tripartite graphs. We show that $π_f(G)<\sqrt 2$ when $G$ is a tree and present a sequence of trees on which the value tends to $4/3$. Finally, we consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let $π^\circ(G)$ be the number of circular orderings needed to separate all pairs and $π_f^\circ(G)$ be the fractional version. Among our results: (1) $π^\circ(G)=1$ if and only $G$ is outerplanar. (2) $π^\circ(G)\le2$ when $G$ is bipartite. (3) $π^\circ(K_n)\ge\log_2\log_3(n-1)$. (4) $π_f^\circ(G)\le\frac{3}{2}$, with equality if and only if $K_4\subseteq G$. (5) $π_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$.

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arXiv2016

Reconstruction from $k$-decks for graphs with maximum degree 2

The $k$-deck of a graph is its multiset of induced subgraphs on $k$ vertices. We prove that $n$-vertex graphs with maximum degree $2$ have the same $k$-decks if each cycle has at least $k+1$ vertices, each path component has at least $k-1$ vertices, and the number of edges is the same. Using this for lower bounds, we obtain for each graph with maximum degree at most $2$ the least $k$ such that it is determined by its $k$-deck. For the $n$-vertex cycle this value is $\lfloor n/2 \rfloor$, and for the $n$-vertex path it is $\lfloor n/2 \rfloor+1$. Also, the least $k$ such that the $k$-deck of an $n$-vertex graph always determines whether it is connected is at least $\lfloor n/2 \rfloor +1$.

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arXiv2016

Spanning Trees in 2-trees

A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood consists of two adjacent vertices. We use this construction order both to inductively list the spanning trees without repetition and to give bounds on the number of them. We determine the $n$-vertex $2$-trees having the most and the fewest spanning trees. The $2$-tree with the fewest is unique; it has $n-2$ vertices of degree $2$ and has $n2^{n-3}$ spanning trees. Those with the most are all those having exactly two vertices of degree $2$, and their number of spanning trees is the Fibonacci number $F_{2n-2}$.

cs.DM↗

arXiv2016

Strongly connectable digraphs and non-transitive dice

We give a new proof of the theorem of Boesch-Tindell and Farzad-Mahdian-Mahmoodian-Saberi-Sadri that a directed graph extends to a strongly connected digraph on the same vertex set if and only if it has no complete directed cut. Our proof bounds the number of edges needed for such an extension; we give examples to demonstrate sharpness. We apply the characterization to a problem on non-transitive dice.

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arXiv2016

The vulnerability of the diameter of enhanced hypercubes

For an interconnection network $G$, the {\it $ω$-wide diameter} $d_ω(G)$ is the least $\ell$ such that any two vertices are joined by $ω$ internally-disjoint paths of length at most $\ell$, and the {\it $(ω-1)$-fault diameter} $D_ω(G)$ is the maximum diameter of a subgraph obtained by deleting fewer than $ω$ vertices of $G$. The enhanced hypercube $Q_{n,k}$ is a variant of the well-known hypercube. Yang, Chang, Pai, and Chan gave an upper bound for $d_{n+1}(Q_{n,k})$ and $D_{n+1}(Q_{n,k})$ and posed the problem of finding the wide diameters and fault diameters of $Q_{n,k}$. By constructing internally disjoint paths between any two vertices in the enhanced hypercube, for $n\ge3$ and $2\le k\le n$ we prove $$ D_ω(Q_{n,k})=d_ω(Q_{n,k})=\begin{cases} d(Q_{n,k}) & \textrm{for $1 \leq ω< n-\lfloor\frac{k}{2}\rfloor$;}\\ d(Q_{n,k})+1 & \textrm{for $n-\lfloor\frac{k}{2}\rfloor \leq ω\leq n+1$.} \end{cases} $$ where $d(Q_{n,k})$ is the diameter of $Q_{n,k}$. These results mean that interconnection networks modelled by enhanced hypercubes are extremely robust.

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arXiv2016

To catch a falling robber

We consider a Cops-and-Robber game played on the subsets of an $n$-set. The robber starts at the full set; the cops start at the empty set. On each turn, the robber moves down one level by discarding an element, and each cop moves up one level by gaining an element. The question is how many cops are needed to ensure catching the robber when the robber reaches the middle level. Aaron Hill posed the problem and provided a lower bound of $2^{n/2}$ for even $n$ and $\binom{n}{\lceil n/2 \rceil}2^{-\lfloor n/2 \rfloor}$ for odd $n$. We prove an upper bound (for all $n$) that is within a factor of $O(\ln n)$ times this lower bound.

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arXiv2015

Coloring, sparseness, and girth

An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-$k$-choosable bipartite graphs, showing that maximum average degree at most $2(k-1)$ is a sharp sufficient condition for $k$-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-$k$-colorable graphs and hypergraphs with any girth $g$.

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arXiv2015

Uniquely cycle-saturated graphs

Given a graph $F$, a graph $G$ is {\it uniquely $F$-saturated} if $F$ is not a subgraph of $G$ and adding any edge of the complement to $G$ completes exactly one copy of $F$. In this paper we study uniquely $C_t$-saturated graphs. We prove the following: (1) a graph is uniquely $C_5$-saturated if and only if it is a friendship graph. (2) There are no uniquely $C_6$-saturated graphs or uniquely $C_7$-saturated graphs. (3) For $t\ge6$, there are only finitely many uniquely $C_t$-saturated graphs (we conjecture that in fact there are none).

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