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Rogers Mathew

Publications and source records attributed to Rogers Mathew.

At least 19 recordsLinked to original sources
arXiv2026

Computational and Combinatorial Results on Conflict-free Choosability

The conflict-free closed neighborhood (CFCN$^*$) chromatic number of a graph $G = (V,E)$ is the smallest positive integer $k$ for which there exists a coloring of a subset of vertices using $k$ colors such that, for every vertex in $V$, there exists a color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) chromatic number is defined analogously. In this paper, we study `list variants' of the above-mentioned coloring parameters. The conflict-free closed neighborhood (CFCN$^*$) choice number of a graph $G = (V,E)$ is the smallest positive integer $k$ such that for every assignment of lists of size $k$ to its vertices, there exists a coloring of a subset of vertices, say $V'$, in which (i) every vertex in $V'$ receives a color from its list, and (ii) for every vertex in $V$ there exists some color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) choice number is defined analogously. D\k{e}bski and Przyby\l o [Journal of Graph Theory, 2022] showed that for any graph $G$ with maximum degree $\Delta$, the CFCN$^*$ chromatic number of its line graph is $O(\ln \Delta)$. This result was later extended to claw-free graphs by Bhyravarapu et al. [Journal of Graph Theory, 2025], who proved that every $K_{1,k}$-free graph $G$ admits a CFCN$^*$ coloring using $O(k\ln \Delta)$ colors. In this paper, we generalize this result to the list setting and show that every $K_{1,k}$-free graph $G$ has a CFCN$^*$ choice number of $O(k\ln \Delta)$. Further, we answer some questions concerning the hardness of computing CFCN$^*$/CFON$^*$ choice numbers posed by Gupta and Mathew [SOFSEM, 2026]; in particular, we show that it is NP-hard to determine whether the CFCN$^*$/CFON$^*$ choice number a graph is equal to $k$, for $k=1,2$.

math.CO↗
arXiv2026

Towards a conjecture on long induced rainbow paths in triangle-free graphs

Given a triangle-free graph $G$ with chromatic number $k$ and a proper vertex coloring $\phi$ of $G$, it is conjectured that $G$ contains an induced rainbow path on $k$ vertices under $\phi$. Scott and Seymour proved the existence of an induced rainbow path on $(\log \log \log k)^{\frac{1}{3}- o(1)}$ vertices. We improve this to $(\log k)^{\frac{1}{2}- o(1)}$ vertices. Further, we prove the existence of an induced path that sees $\frac{k}{2}$ colors.

math.CO↗
arXiv2024

Bounds and Hardness Results for Conflict-free Choosability

A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other vertex in that hyperedge. The minimum number of colors required for such a coloring is known as the '(partial) conflict-free chromatic number' of $\mathcal{H}$. It is easy to see that the conflict-free chromatic number of a hypergraph is at most its partial conflict-free chromatic number plus one. Conflict-free coloring has also been studied on the open/closed neighborhood hypergraphs of a given graph under the name open/closed neighborhood conflict-free coloring. In this paper, we study partial and full list variants of conflict-free coloring where, for every vertex $v$, we are given a list of admissible colors $L_v$ such that $v$ is allowed to be colored only from $L_v$. Bhyravarapu, Kalyanasundaram, and Mathew [Journal of Graph Theory, 2021] showed that the closed-neighborhood conflict-free chromatic number of any graph $G$ with maximum degree $\Delta$ is at most $O(\ln^2 \Delta)$. In this paper, we extend the $O(\ln^2 \Delta)$ upper bound to the partial list variant of the closed-neighborhood conflict-free chromatic number. Further, we establish computational complexity results concerning the list open/closed-neighborhood conflict-free chromatic numbers.

math.CO↗
arXiv2023

Extremal Results on Conflict-free Coloring

A conflict-free open neighborhood coloring of a graph is an assignment of colors to the vertices such that for every vertex there is a color that appears exactly once in its open neighborhood. For a graph $G$, the smallest number of colors required for such a coloring is called the conflict-free open neighborhood (CFON) chromatic number and is denoted by $\chi_{ON}(G)$. By considering closed neighborhood instead of open neighborhood, we obtain the analogous notions of conflict-free closed neighborhood (CFCN) coloring, and CFCN chromatic number (denoted by $\chi_{CN}(G)$). The notion of conflict-free coloring was introduced in 2002, and has since received considerable attention. In this paper, we study some extremal questions related to CFON and CFCN coloring.

math.CO↗
arXiv2022

Flexible list colorings: Maximizing the number of requests satisfied

Flexible list coloring was introduced by Dvo\v{r}\'{a}k, Norin, and Postle in 2019. Suppose $0 \leq \epsilon \leq 1$, $G$ is a graph, $L$ is a list assignment for $G$, and $r$ is a function with non-empty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$ ($r$ is called a request of $L$). The triple $(G,L,r)$ is $\epsilon$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $\epsilon|D|$ vertices in $D$. We say $G$ is $(k, \epsilon)$-flexible if $(G,L',r')$ is $\epsilon$-satisfiable whenever $L'$ is a $k$-assignment for $G$ and $r'$ is a request of $L'$. It was shown by Dvo\v{r}\'{a}k et al. that if $d+1$ is prime, $G$ is a $d$-degenerate graph, and $r$ is a request for $G$ with domain of size $1$, then $(G,L,r)$ is $1$-satisfiable whenever $L$ is a $(d+1)$-assignment. In this paper, we extend this result to all $d$ for bipartite $d$-degenerate graphs. The literature on flexible list coloring tends to focus on showing that for a fixed graph $G$ and $k \in \mathbb{N}$ there exists an $\epsilon > 0$ such that $G$ is $(k, \epsilon)$-flexible, but it is natural to try to find the largest possible $\epsilon$ for which $G$ is $(k,\epsilon)$-flexible. In this vein, we improve a result of Dvo\v{r}\'{a}k et al., by showing $d$-degenerate graphs are $(d+2, 1/2^{d+1})$-flexible. In pursuit of the largest $\epsilon$ for which a graph is $(k,\epsilon)$-flexible, we observe that a graph $G$ is not $(k, \epsilon)$-flexible for any $k$ if and only if $\epsilon > 1/ \rho(G)$, where $\rho(G)$ is the Hall ratio of $G$, and we initiate the study of the list flexibility number of a graph $G$, which is the smallest $k$ such that $G$ is $(k,1/ \rho(G))$-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.

math.CO↗
arXiv2022

On hierarchically closed fractional intersecting families

For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1, \dotsc, A_t \in \mathcal{F}$ there exists $\theta \in L$ such that $\lvert A_1 \cap \dotsb \cap A_t \rvert \in \{ \theta \lvert A_1 \rvert, \dotsc, \theta \lvert A_t \rvert\}$. In this paper we show that for $r \geq 3$ and $L = \{\theta\}$ any fractional $r$-closed $\theta$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $\theta = 1/2$ we have a tight upper bound of $\lfloor \frac{3n}{2} \rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism.

math.CO↗
arXiv2022

Bounding threshold dimension: realizing graphic Boolean functions as the AND of majority gates

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic vectors of the cliques of $G$. Introduced in [Chv{\'a}tal and Hammer, Annals of Discrete Mathematics, 1977], the \emph{threshold dimension} of a graph $G$, denoted by $\dimth(G)$, is the minimum number of threshold graphs whose intersection yields $G$. Given a graph $G$ on $n$ vertices, in line with Chv{\'a}tal and Hammer, $f_G\colon \{0,1\}^n \rightarrow \{0,1\}$ is the Boolean function that has the property that $f_G(x) = 1$ if and only if $x$ is the characteristic vector of a clique in $G$. A Boolean function $f$ for which there exists a graph $G$ such that $f=f_G$ is called a \emph{graphic} Boolean function. It follows that for a graph $G$, $\dimth(G)$ is precisely the minimum number of \emph{majority} gates whose AND (or conjunction) realizes the graphic Boolean function $f_G$. The fact that there exist Boolean functions which can be realized as the AND of only exponentially many majority gates motivates us to study threshold dimension of graphs. We give tight or nearly tight upper bounds for the threshold dimension of a graph in terms of its treewidth, maximum degree, degeneracy, number of vertices, size of a minimum vertex cover, etc. We also study threshold dimension of random graphs and graphs with high girth.

math.CO↗
arXiv2021

Conflict-free coloring on open neighborhoods of claw-free graphs

The `Conflict-Free Open (Closed) Neighborhood coloring', abbreviated CFON (CFCN) coloring, of a graph $G$ using $r$ colors is a coloring of the vertices of $G$ such that every vertex sees some color exactly once in its open (closed) neighborhood. The minimum $r$ such that $G$ has a CFON (CFCN) coloring using $r$ colors is called the `CFON chromatic number' (`CFCN chromatic number') of $G$. This is denoted by $\chi_{CF}^{ON}(G)$ ($\chi_{CF}^{CN}(G)$). D\k ebski and Przyby\l{}o in [J. Graph Theory, 2021] showed that if $G$ is a line graph with maximum degree $\Delta$, then $\chi_{CF}^{CN}(G) = O(\ln \Delta)$. As an open question, they asked if the result could be extended to claw-free ($K_{1,3}$-free) graphs, which are a superclass of line graphs. For $k\geq 3$, we show that if $G$ is $K_{1,k}$-free, then $\chi_{CF}^{ON}(G) = O(k^2\ln \Delta)$. Since it is known that the CFCN chromatic number of a graph is at most twice its CFON chromatic number, this answers the question posed by D\k{e}bski and Przyby\l{}o.

math.CO↗
arXiv2021

Pliable Index Coding via Conflict-Free Colorings of Hypergraphs

In the pliable index coding (PICOD) problem, a server is to serve multiple clients, each of which possesses a unique subset of the complete message set as side information and requests a new message which it does not have. The goal of the server is to do this using as few transmissions as possible. This work presents a hypergraph coloring approach to the scalar PICOD problem. A \textit{conflict-free coloring} of a hypergraph is known from literature as an assignment of colors to its vertices so that each hyperedge of the graph contains one uniquely colored vertex. For a given PICOD problem represented by a hypergraph consisting of messages as vertices and request-sets as hyperedges, we present achievable PICOD schemes using conflict-free colorings of the PICOD hypergraph. Various graph theoretic parameters arising out of such colorings (and some new coloring variants) then give a number of upper bounds on the optimal PICOD length, which we study in this work. Suppose the PICOD hypergraph has $m$ vertices and $n$ hyperedges, where every hyperedge overlaps with at most $\Gamma$ other hyperedges. We show easy to implement randomized algorithms for the following: (a) For the single request case, we give a PICOD of length $O(\log^2\Gamma)$. This result improves over known achievability results for some parameter ranges, (b) For the $t$-request case, we give an MDS code of length $\max(O(\log \Gamma \log m), O(t \log m))$. Further if the hyperedges (request sets) are sufficiently large, we give a PICOD of the same length as above, which is not based on MDS construction. In general, this gives an improvement over prior achievability results. Our codes are of near-optimal length (up to a multiplicative factor of $\log t$).

cs.IT↗
arXiv2020

Conflict-Free Coloring of Star-Free Graphs on Open Neighborhoods

Given a graph, the conflict-free coloring problem on open neighborhoods (CFON) asks to color the vertices of the graph so that all the vertices have a uniquely colored vertex in its open neighborhood. The smallest number of colors required for such a coloring is called the conflict-free chromatic number and denoted $\chi_{ON}(G)$. In this note, we study this problem on $S_k$-free graphs where $S_k$ is a star on $k+1$ vertices. When $G$ is $S_k$-free, we show that $\chi_{ON}(G) = O(k\cdot \log^{2+\epsilon}\Delta)$, for any $\epsilon > 0$, where $\Delta$ denotes the maximum degree of $G$. Further, we show existence of claw-free ($S_3$-free) graphs that require $\Omega(\log \Delta)$ colors.

math.CO↗
arXiv2020

Modular and fractional L-intersecting families of vector spaces

In the first part of this paper, we prove a theorem which is the $q$-analogue of a generalized modular Ray-Chaudhuri-Wilson Theorem shown in [Alon, Babai, Suzuki, J. Combin. Theory Series A, 1991]. It is also a generalization of the main theorem in [Frankl and Graham, European J. Combin. 1985] under certain circumstances. In the second part of this paper, we prove $q$-analogues of results on a recent notion called \emph{fractional $L$-intersecting family} for families of subspaces of a given vector space. We use the above theorem to obtain a general upper bound to the cardinality of such families. We give an improvement to this general upper bound in certain special cases.

math.CO↗
arXiv2020

Conflict-free coloring on closed neighborhoods of bounded degree graphs

The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by $\chi_{CN}(G)$, is the minimum number of colors required to color the vertices of $G$ such that for every vertex, there is a color that appears exactly once in its closed neighborhood. Pach and Tardos [Combin. Probab. Comput. 2009] showed that $\chi_{CN}(G) = O(\log^{2+\varepsilon} \Delta)$, for any $\varepsilon > 0$, where $\Delta$ is the maximum degree. In [Combin. Probab. Comput. 2014], Glebov, Szab\'o and Tardos showed existence of graphs $G$ with $\chi_{CN}(G) = \Omega(\log^2\Delta)$. In this paper, we bridge the gap between the two bounds by showing that $\chi_{CN}(G) = O(\log^2 \Delta)$.

math.CO↗
arXiv2019

Hamiltonian and Pseudo-Hamiltonian Cycles and Fillings In Simplicial Complexes

We introduce and study a $d$-dimensional generalization of Hamiltonian cycles in graphs - the Hamiltonian $d$-cycles in $K_n^d$ (the complete simplicial $d$-complex over a vertex set of size $n$). Those are the simple $d$-cycles of a complete rank, or, equivalently, of size $1 + {{n-1} \choose d}$. The discussion is restricted to the fields $F_2$ and $Q$. For $d=2$, we characterize the $n$'s for which Hamiltonian $2$-cycles exist. For $d=3$ it is shown that Hamiltonian $3$-cycles exist for infinitely many $n$'s. In general, it is shown that there always exist simple $d$-cycles of size ${{n-1} \choose d} - O(n^{d-3})$. All the above results are constructive. Our approach naturally extends to (and in fact, involves) $d$-fillings, generalizing the notion of $T$-joins in graphs. Given a $(d-1)$-cycle $Z^{d-1} \in K_n^d$, ~$F$ is its $d$-filling if $\partial F = Z^{d-1}$. We call a $d$-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size ${{n-1} \choose d}$. If a Hamiltonian $d$-cycle $Z$ over $F_2$ contains a $d$-simplex $\sigma$, then $Z\setminus \sigma$ is a a Hamiltonian $d$-filling of $\partial \sigma$ (a closely related fact is also true for cycles over $Q$). Thus, the two notions are closely related. Most of the above results about Hamiltonian $d$-cycles hold for Hamiltonian $d$-fillings as well.

math.CO↗
arXiv2019

Fractional cross intersecting families

Let $\mathcal{A}=\{A_{1},...,A_{p}\}$ and $\mathcal{B}=\{B_{1},...,B_{q}\}$ be two families of subsets of $[n]$ such that for every $i\in [p]$ and $j\in [q]$, $|A_{i}\cap B_{j}|= \frac{c}{d}|B_{j}|$, where $\frac{c}{d}\in [0,1]$ is an irreducible fraction. We call such families "$\frac{c}{d}$-cross intersecting families". In this paper, we find a tight upper bound for the product $|\mathcal{A}||\mathcal{B}|$ and characterize the cases when this bound is achieved for $\frac{c}{d}=\frac{1}{2}$. Also, we find a tight upper bound on $|\mathcal{A}||\mathcal{B}|$ when $\mathcal{B}$ is $k$-uniform and characterize, for all $\frac{c}{d}$, the cases when this bound is achieved.

math.CO↗
arXiv2018

Target Set Selection parameterized by vertex cover and more

Given a simple, undirected graph $G$ with a threshold function $\tau:V(G) \rightarrow \mathbb{N}$, the \textsc{Target Set Selection} (TSS) problem is about choosing a minimum cardinality set, say $S \subseteq V(G)$, such that starting a diffusion process with $S$ as its seed set will eventually result in activating all the nodes in $G$. For any non-negative integer $i$, we say a set $T\subseteq V(G)$ is a "degree-$i$ modulator" of $G$ if the degree of any vertex in the graph $G-T$ is at most $i$. Degree-$0$ modulators of a graph are precisely its vertex covers. Consider a graph $G$ on $n$ vertices and $m$ edges. We have the following results on the TSS problem: -> It was shown by Nichterlein et al. [Social Network Analysis and Mining, 2013] that it is possible to compute an optimal-sized target set in $O(2^{(2^{t}+1)t}\cdot m)$ time, where $t$ denotes the cardinality of a minimum degree-$0$ modulator of $G$. We improve this result by designing an algorithm running in time $2^{O(t\log t)}n^{O(1)}$. -> We design a $2^{2^{O(t)}}n^{O(1)}$ time algorithm to compute an optimal target set for $G$, where $t$ is the size of a minimum degree-$1$ modulator of $G$.

cs.CC↗
arXiv2018

Local Boxicity and Maximum Degree

The \emph{local boxicity} of a graph $G$, denoted by $lbox(G)$, is the minimum positive integer $l$ such that $G$ can be obtained using the intersection of $k$ (, where $k \geq l$,) interval graphs where each vertex of $G$ appears as a non-universal vertex in at most $l$ of these interval graphs. Let $G$ be a graph on $n$ vertices having $m$ edges. Let $\Delta$ denote the maximum degree of a vertex in $G$. We show that, (i) $lbox(G) \leq 2^{13\log^{*}{\Delta}} \Delta$. There exist graphs of maximum degree $\Delta$ having a local boxicity of $\Omega(\frac{\Delta}{\log\Delta})$. (ii) $lbox(G) \in O(\frac{n}{\log{n}})$. There exist graphs on $n$ vertices having a local boxicity of $\Omega(\frac{n}{\log n})$. (iii) $lbox(G) \leq (2^{13\log^{*}{\sqrt{m}}} + 2 )\sqrt{m}$. There exist graphs with $m$ edges having a local boxicity of $\Omega(\frac{\sqrt{m}}{\log m})$. (iv) the local boxicity of $G$ is at most its \emph{product dimension}. This connection helps us in showing that the local boxicity of the \emph{Kneser graph} $K(n,k)$ is at most $\frac{k}{2} \log{\log{n}}$. The above results can be extended to the \emph{local dimension} of a partially ordered set due to the known connection between local boxicity and local dimension. Finally, we show that the \emph{cubicity} of a graph on $n$ vertices of girth greater than $g+1$ is $O(n^{\frac{1}{\lfloor g/2\rfloor}}\log n)$.

math.CO↗
arXiv2018

New bounds on the anti-Ramsey numbers of star graphs

The anti-Ramsey number $ar(G,H)$ with input graph $G$ and pattern graph $H$, is the maximum positive integer $k$ such that there exists an edge coloring of $G$ using $k$ colors, in which there are no rainbow subgraphs isomorphic to $H$ in $G$. ($H$ is rainbow if all its edges get distinct colors). The concept of anti-Ramsey number was introduced by Erd\"os, Simanovitz, and S\'os in 1973. Thereafter several researchers investigated this concept in the combinatorial setting. Recently, Feng et al. revisited the anti-Ramsey problem for the pattern graph $K_{1,t}$ (for $t \geq 3$) purely from an algorithmic point of view due to its applications in interference modeling of wireless networks. They posed it as an optimization problem, the maximum edge $q$-coloring problem. For a graph $G$ and an integer $q\geq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of colors in an edge $q$-coloring of the graph $G$. Note that the optimum value of the edge $q$-coloring problem of $G$ equals $ar(G,K_{1,q+1})$. In this paper, we study $ar(G,K_{1,t})$, the anti-Ramsey number of stars, for each fixed integer $t\geq 3$, both from combinatorial and algorithmic point of view. The first of our main results presents an upper bound for $ar(G,K_{1,q+1})$, in terms of number of vertices and the minimum degree of $G$. The second one improves this result for the case of triangle-free input graphs. For a positive integer $t$, let $H_t$ denote a subgraph of $G$ with maximum number of possible edges and maximum degree $t$. Our third main result presents an upper bound for $ar(G,K_{1,q+1})$ in terms of $|E(H_{q-1})|$. All our results have algorithmic consequences.

cs.DM↗