Some results on multithreshold graphs

Jamison and Sprague defined a graph $G$ to be a $k$-threshold graph with thresholds $\theta_1 , \ldots, \theta_k$ (strictly increasing) if one can assign real numbers $(r_v)_{v \in V(G)}$, called ranks, such that for every pair of vertices $v,w$, we have $vw \in E(G)$ if and only if the inequality $\theta_i \leq r_v + r_w$ holds for an odd number of indices $i$. When $k=1$ or $k=2$, the precise choice of thresholds $\theta_1, \ldots, \theta_k$ does not matter, as a suitable transformation of the ranks transforms a representation with one choice of thresholds into a representation with any other choice of thresholds. Jamison asked whether this remained true for $k \geq 3$ or whether different thresholds define different classes of graphs for such $k$, offering \$50 for a solution of the problem. Letting$C_t$for$t > 1$denote the class of$3$-threshold graphs with thresholds$-1, 1, t$, we prove that there are infinitely many distinct classes$C_t$, answering Jamison's question. We also consider some other problems on multithreshold graphs, some of which remain open.Comment: 6 pages, 1 figur

Graphs with α1\alpha_1 and τ1\tau_1 both large

Given a graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of an edge set containing at most one edge from each triangle of $G$. Erd\H{o}s, Gallai, and Tuza introduced several problems with the unifying theme that $\alpha_1(G)$ and $\tau_1(G)$ cannot both be "very large"; the most well-known such problem is their conjecture that $\alpha_1(G) + \tau_1(G) \leq |V(G)|^2/4$, which was proved by Norin and Sun. We consider three other problems within this theme (two introduced by Erd\H{o}s, Gallai, and Tuza, another by Norin and Sun), all of which request an upper bound either on $\min\{\alpha_1(G), \tau_1(G)\}$ or on $\alpha_1(G) + k\tau_1(G)$ for some constant $k$, and prove the existence of graphs for which these quantities are "large".Comment: 6 pages; improved exposition a bit and fixed an issue regarding integrality from the earlier versio

Complexity of a Disjoint Matching Problem on Bipartite Graphs

We consider the following question: given an $(X,Y)$-bigraph $G$ and a set $S \subset X$, does $G$ contain two disjoint matchings $M_1$ and $M_2$ such that $M_1$ saturates $X$ and $M_2$ saturates $S$? When $|S|\geq |X|-1$, this question is solvable by finding an appropriate factor of the graph. In contrast, we show that when $S$ is allowed to be an arbitrary subset of $X$, the problem is NP-hard.Comment: 6 pages, 1 figur

Extremal Aspects of the Erd\H{o}s--Gallai--Tuza Conjecture

Erd\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $\alpha_1(G) + \tau_1(G) \leq n^2/4$? We also consider a variant on this conjecture: if $\tau_B(G)$ is the smallest size of an edge set whose deletion makes $G$ bipartite, does the stronger inequality $\alpha_1(G) + \tau_B(G) \leq n^2/4$ always hold? By considering the structure of a minimal counterexample to each version of the conjecture, we obtain two main results. Our first result states that any minimum counterexample to the original Erd\H{o}s--Gallai--Tuza Conjecture has "dense edge cuts", and in particular has minimum degree greater than $n/2$. This implies that the conjecture holds for all graphs if and only if it holds for all triangular graphs (graphs where every edge lies in a triangle). Our second result states that $\alpha_1(G) + \tau_B(G) \leq n^2/4$ whenever $G$ has no induced subgraph isomorphic to $K_4^-$, the graph obtained from the complete graph $K_4$ by deleting an edge. Thus, the original conjecture also holds for such graphs.Comment: 5 pages. Updated with journal reference, expanded background, and a few other minor change

On a Conjecture of Erd\H{o}s, Gallai, and Tuza

Erd\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $\alpha_1(G) + \tau_1(G) \leq n^2/4$? We have two main results. We first obtain the upper bound $\alpha_1(G) + \tau_1(G) \leq 5n^2/16$, as a partial result towards the Erd\H{o}s--Gallai--Tuza conjecture. We also show that always $\alpha_1(G) \leq n^2/2 - m$, where $m$ is the number of edges in $G$; this bound is sharp in several notable cases.Comment: 5 pages, minor revisions: added new details, new conjecture, and cleaned up notation slightl

Favaron's Theorem, k-dependence, and Tuza's Conjecture

A vertex set $D$ in a graph $G$ is $k$-dependent if $G[D]$ has maximum degree at most $k-1$, and $k$-dominating if every vertex outside $D$ has at least $k$ neighbors in $D$. Favaron proved that if $D$ is a $k$-dependent set maximizing the quantity $k|D| - |E(G[D])|$, then $D$ is $k$-dominating. We extend this result, showing that such sets satisfy a stronger structural property, and we find a surprising connection between Favaron's theorem and a conjecture of Tuza regarding packing and covering of triangles.Comment: 12 pages. Strengthened main theorem and simplified its proof by replacing vertex-orderings with orientation

Maximal kk-Edge-Colorable Subgraphs, Vizing's Theorem, and Tuza's Conjecture

We prove that if $M$ is a maximal $k$-edge-colorable subgraph of a multigraph $G$ and if $F = \{v \in V(G) : d_M(v) \leq k-\mu(v)\}$, then $d_F(v) \leq d_M(v)$ for all $v \in F$. (When $G$ is a simple graph, the set $F$ is just the set of vertices having degree less than $k$ in $M$.) This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.Comment: 11 pages, 1 figure. Fixed some inaccurate references to "Vizing's Theorem" (the stronger version cited here is in fact due to Ore), cleared up some muddled results in the section about forests, simplified some notation, and made other various readability improvement

tt-cores for (Δ+t)(\Delta+t)-edge-colouring

We extend the edge-coloring notion of core (subgraph induced by the vertices of maximum degree) to $t$-core (subgraph induced by the vertices $v$ with $d(v)+\mu(v)> \Delta+t$), and find a sufficient condition for $(\Delta+t)$-edge-coloring. In particular, we show that for any $t\geq 0$, if the $t$-core of $G$ has multiplicity at most $t+1$, with its edges of multiplicity $t+1$ inducing a multiforest, then $\chi'(G) \leq \Delta+t$. This extends previous work of Ore, Fournier, and Berge and Fournier. A stronger version of our result (which replaces the multiforest condition with a vertex-ordering condition) generalizes a theorem of Hoffman and Rodger about cores of $\Delta$-edge-colourable simple graphs. In fact, our bounds hold not only for chromatic index, but for the \emph{fan number} of a graph, a parameter introduced by Scheide and Stiebitz as an upper bound on chromatic index. We are able to give an exact characterization of the graphs $H$ such that $\mathrm{Fan}(G) \leq \Delta(G)+t$ whenever $G$ has $H$ as its $t$-core.Comment: 15 pages, 2 figures. This version fixes an issue with the definition of the fan number, and makes several smaller improvement

Environmental Evolutionary Graph Theory

Understanding the influence of an environment on the evolution of its resident population is a major challenge in evolutionary biology. Great progress has been made in homogeneous population structures while heterogeneous structures have received relatively less attention. Here we present a structured population model where different individuals are best suited to different regions of their environment. The underlying structure is a graph: individuals occupy vertices, which are connected by edges. If an individual is suited for their vertex, they receive an increase in fecundity. This framework allows attention to be restricted to the spatial arrangement of suitable habitat. We prove some basic properties of this model and find some counter-intuitive results. Notably, 1) the arrangement of suitable sites is as important as their proportion, and, 2) decreasing the proportion of suitable sites may result in a decrease in the fixation time of an allele

Correlation Clustering and Biclustering with Locally Bounded Errors

We consider a generalized version of the correlation clustering problem, defined as follows. Given a complete graph $G$ whose edges are labeled with $+$ or $-$, we wish to partition the graph into clusters while trying to avoid errors: $+$ edges between clusters or $-$ edges within clusters. Classically, one seeks to minimize the total number of such errors. We introduce a new framework that allows the objective to be a more general function of the number of errors at each vertex (for example, we may wish to minimize the number of errors at the worst vertex) and provide a rounding algorithm which converts "fractional clusterings" into discrete clusterings while causing only a constant-factor blowup in the number of errors at each vertex. This rounding algorithm yields constant-factor approximation algorithms for the discrete problem under a wide variety of objective functions.Comment: 20 pages, reorganized paper to emphasize the key properties of the rounding algorithm and the broader class of possible objective function