Shashanka Kulamarva

Jesper Jansson, Shashanka Kulamarva, Yukihiro Murakami et al.

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 whose 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 explicit bound building on the results of Bastide et al., and identify the ranges of $k$ and $n$ for which our bound outperforms theirs. 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.

Nevil Anto, Manu Basavaraju, Shashanka Kulamarva

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 $Δ$.

Dhanyamol Antony, L. Sunil Chandran, Anita Das et al.

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 $d \geq 4$.

Dhanyamol Antony, L. Sunil Chandran, Anita Das et al.

Graph burning is a discrete-time process that models the propagation of information in a network. Initially, we have an undirected graph of unburned vertices. At each time step, an unburned vertex is chosen to burn; additionally, unburned vertices with at least one burned neighbor from the previous step also become burned. Once a vertex is burned, it remains burned for all future steps. The burning number of a graph is the minimum number of steps to burn all the vertices of the graph. The BURNING NUMBER PROBLEM asks whether the burning number of an input graph $G$ is at most $k$ or not. In this paper, we study the BURNING NUMBER PROBLEM both from an algorithmic and a structural viewpoint. The BURNING NUMBER PROBLEM is known to be NP-complete for interval graphs. Here, we prove that this problem is NP-complete even when restricted to connected proper interval graphs. The well-known burning number conjecture asserts that the burning number of a connected graph of order $n$ is at most $\lceil \sqrt{n}~\rceil$. In line with this conjecture, the upper and lower bounds of the burning number are well-studied for various graph classes. Here, we provide an improved upper bound for the burning number of connected $P_k$-free graphs and show that the 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 their relationship with the classical burning and evaluate the algorithmic complexity of these variants.