arXiv subjects

Shashanka Kulamarva

Publications and source records attributed to Shashanka Kulamarva.

9 recordsLinked to original sources

arXiv2026

Hardness of Burning Number Problem on Regular Graphs

The Burning Number Problem (BNP) models the spread of information or contagion in a network through a discrete-time process on a graph. At each step, one new vertex is selected as a burning source, while fire simultaneously spreads from previously burned vertices to their neighbors. The burning number of a graph is the minimum number of steps required to burn all vertices. The decision version asks whether the burning number is at most a given integer $k$. BNP is known to be NP-complete even on restricted graph classes such as path forests. We study BNP on connected regular graphs, a natural and previously unexplored graph class. We prove that BNP is NP-complete on connected cubic graphs, and moreover APX-hard under this restriction. We further show that BNP remains APX-hard on connected $d$-regular graphs for every fixed $d \geq 4$.

cs.DS↗

arXiv2026

Burning Graph Powers and Branching Trees

Graph burning is a discrete-time process that models the spread of social contagion. Initially, all vertices are unburned. In each round, one unburned vertex is selected and burned, while any unburned vertex that has a burned neighbour from the previous round also becomes burned. The burning number of a graph is the minimum number of rounds needed to burn the entire graph. In this paper, we study the burning number of graph powers. First, we show that for a connected graph~$G$, its graph power~$G^k$ contains a~$(k+1)^+$-branching tree as a spanning tree. A~$(k+1)^+$-branching tree is one in which all internal vertices have degree at least~$k+1$. We then show that $(k+1)^+$-branching trees on~$n$ vertices have burning number at most $\left\lceil{\sqrt{\frac{4(k-1)n}{k^2}}}~\right\rceil$. As the burning number of a graph is at most the burning number of any of its spanning trees, this gives an upper bound on the burning number of graph powers. We also derive an alternative upper bound on the burning number of~$k^+$-branching trees using the strongest currently known general burning number bound [Bastide et al.]. We then identify the ranges of~$k$ and~$n$ for which our bound outperforms or matches this alternative bound. Finally, we show that~$b(G^k) \le (1+o(1))\sqrt{n/k}$ based on the asymptotic burning number bound of Norin and Turcotte.

math.CO↗

arXiv2025

A Graph-Theoretic Model for a Generic Three-Jug Puzzle

A classic three-jug puzzle asks, given three jugs $A$, $B$, and $C$ with fixed maximum capacities, with jug $A$ filled with wine to its maximum capacity, whether is it possible to divide the wine into two halves by pouring it from one jug to another without using any other measuring devices. However, we consider a generic version of the three-jug puzzle and present an independent graph-theoretic model to determine whether the puzzle has a solution at all. If it has a solution, then the same can be determined using this model. We also present the sketch of an algorithm to determine the solution of the puzzle.

math.HO↗

arXiv2025

Acyclic Edge Coloring of 3-sparse Graphs

A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamč\'ık conjectured that for a graph $G$ with maximum degree $Δ$, $a'(G) \le Δ+2$. A graph $G$ is said to be $3$-sparse if every edge in $G$ is incident on at least one vertex of degree at most $3$. We prove the conjecture for the class of $3$-sparse graphs. Further, we give a stronger bound of $Δ+1$, if there exists an edge $xy$ in the graph with $d_G(x)+ d_G(y) < Δ+3$. When $ Δ> 3$, the $3$-sparse graphs where no such edge exists is the set of bipartite graphs where one partition has vertices with degree exactly $3$ and the other partition has vertices with degree exactly $Δ$.

math.CO↗

arXiv2024

Graph Burning: Bounds and Hardness

Graph burning is a discrete-time process that models the propagation of information in a network. Given an undirected graph whose vertices are initially unburned, the process evolves in discrete rounds. At each round, an unburned vertex is selected and burned, while any unburned vertex adjacent to a vertex burned in the previous round also becomes burned. The burning number of a graph is the minimum number of steps to burn all its vertices. The BURNING NUMBER PROBLEM asks whether the burning number of an input graph $G$ is at most $k$. In this paper, we investigate the graph burning problem from both algorithmic and structural viewpoints. Although the problem is known to be NP-complete on interval graphs, we strengthen this result by proving that it remains NP-complete even when restricted to connected proper interval graphs. We also study the burning number of $P_k$-free graphs. Motivated by the well-known burning number conjecture, which states that every connected graph of order $n$ has burning number at most $\lceil \sqrt{n}~\rceil$, we establish an improved upper bound for connected $P_k$-free graphs and show that this bound is tight up to an additive constant of $1$. Finally, we study two variants of the problem: edge burning and total burning. We establish fundamental relationships between these variants and the classical burning, and we determine the computational complexity of the corresponding decision problems.

math.CO↗

arXiv2024

An improved upper bound for the domination number of a graph

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $γ(G) \le \lfloor\sqrt{n}\rfloor$ which gives a substantially improved upper bound. In this paper, we give a condition necessary for a graph $G$ to have $γ(G) \le \lfloor\sqrt{n}\rfloor$, and some conditions sufficient for a graph $G$ to have $γ(G) \le \lfloor\sqrt{n}\rfloor$. We also present a characterization of all connected graphs $G$ of order $n$ with $γ(G) = \lfloor\sqrt{n}\rfloor$. Further, we prove that for a graph $G$ not satisfying $rad(G)=diam(G)=rad(\overline{G})=diam(\overline{G})=2$, deciding whether $γ(G) \le \lfloor\sqrt{n}\rfloor$ or $γ(\overline{G}) \le \lfloor\sqrt{n}\rfloor$ can be done in polynomial time. We conjecture that this decision problem can be solved in polynomial time for any graph $G$.

math.CO↗

arXiv2023

Spanning caterpillar in biconvex bipartite graphs

A bipartite graph $G=(A, B, E)$ is said to be a biconvex bipartite graph if there exist orderings $<_A$ in $A$ and $<_B$ in $B$ such that the neighbors of every vertex in $A$ are consecutive with respect to $<_B$ and the neighbors of every vertex in $B$ are consecutive with respect to $<_A$. A caterpillar is a tree that will result in a path upon deletion of all the leaves. In this note, we prove that there exists a spanning caterpillar in any connected biconvex bipartite graph. Besides being interesting on its own, this structural result has other consequences. For instance, this directly resolves the burning number conjecture for biconvex bipartite graphs.

math.CO↗

arXiv2023

Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs

An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamč\'ık conjectured that $a'(G) \le Δ+2$ for any graph $G$ with maximum degree $Δ$. A graph $G$ is said to be $k$-degenerate if every subgraph of $G$ has a vertex of degree at most $k$. Basavaraju and Chandran proved that the conjecture is true for $2$-degenerate graphs. We prove that for a $3$-degenerate graph $G$, $a'(G) \le Δ+5$, thereby bringing the upper bound closer to the conjectured bound. We also consider $k$-degenerate graphs with $k \ge 4$ and give an upper bound for the acyclic chromatic index of the same.

math.CO↗

arXiv2023

Acyclic Chromatic Index of Chordless Graphs

An acyclic edge coloring of a graph is a proper edge coloring in which there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum positive integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. It has been conjectured by Fiamč\'ık that $a'(G) \le Δ+2$ for any graph $G$ with maximum degree $Δ$. Linear arboricity of a graph $G$, denoted by $la(G)$, is the minimum number of linear forests into which the edges of $G$ can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. Every $2$-connected chordless graph is a minimally $2$-connected graph. It was shown by Basavaraju and Chandran that if $G$ is $2$-degenerate, then $a'(G) \le Δ+1$. Since chordless graphs are also $2$-degenerate, we have $a'(G) \le Δ+1$ for any chordless graph $G$. Machado, de Figueiredo and Trotignon proved that the chromatic index of a chordless graph is $Δ$ when $Δ\ge 3$. They also obtained a polynomial time algorithm to color a chordless graph optimally. We improve this result by proving that the acyclic chromatic index of a chordless graph is $Δ$, except when $Δ=2$ and the graph has a cycle, in which case it is $Δ+1$. We also provide the sketch of a polynomial time algorithm for an optimal acyclic edge coloring of a chordless graph. As a byproduct, we also prove that $la(G) = \lceil \frac{Δ}{2} \rceil$, unless $G$ has a cycle with $Δ=2$, in which case $la(G) = \lceil \frac{Δ+1}{2} \rceil = 2$. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado, de Figueiredo and Trotignon for this class of graphs. This might be of independent interest.

math.CO↗