arXiv subjects

Manoj Belavadi

Publications and source records attributed to Manoj Belavadi.

7 recordsLinked to original sources

arXiv2026

Vertex-critical co-gem-free graphs

Given a graph $G$, let $χ(G)$ denote the chromatic number of $G$. For $k\in \mathbb{N}$, a graph $G$ is $k$-$vertex$-$critical$ if $χ(G)=k$ and $χ(G-v)< k$ for all $v\in V(G)$. A recent problem of Beaton and Cameron [TCS 1042 (2025) 115234] asks for which graphs $H$ of order five, are there finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, for all $k\in \mathbb{N}$? Here we identify three distinct graphs on five vertices that yield an affirmative answer to this problem. More precisely, we show that for each $k\in \mathbb{N}$, there are finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, where $H\in \{$paraglider, dart, house$\}$, by analysing the structure of such graphs. Our results together with a result of Couturier et al. [Algorithmica 71:1 (2015) 21--35] imply that for each $k\in \mathbb{N}$, there is a polynomial-time certifying algorithm for $k$-COLORING of (co-gem, $H$)-free graphs, where $H\in \{$paraglider, dart, house$\}$.

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arXiv2026

Structural description of (bull, house)-free graphs

The bull is a graph consisting of a triangle and two pendant edges. The P_5 is the chordless path on five vertices. The house is the complement of a P_5. A graph is k-critical if it is k-chromatic but each of its proper induced subgraphs is (k-1)-colorable. It is known that the number of k-critical P_5-free graphs and bull-free graphs are infinite for large enough k. We give a structural description of (bull, house)-free graphs and also (bull, P_5)-free graphs. Using these structural properties we prove that for any fixed k, the number of k-critical (bull, P_5)-free graphs is finite. This improves on a result of Huang, Li and Xia (Critical (P_5, bull)-free graphs, Discrete Applied Mathematics 334 (2023) 15-25). A graph G is perfectly divisible if for each induced subgraph H of G with at least one edge, V(H) can be partitioned into two sets V_1, V_2 such that every largest clique of H contains a vertex in V_i for i = 1,2. Chudnovsky and Sivaraman proved that (P_5, bull)-free graphs are perfectly divisible (Perfect divisibility and 2-divisibility, Journal of Graph Theory 90 (2019) 54-60). Our structural result allows us to give a short proof of this theorem.

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arXiv2025

Kempe changes in $H$-free graphs

Given a $k$-colouring of a graph $G$ and two of the colours, a $Kempe$ $chain$ is a connected component of the subgraph of $G$ induced by the vertices coloured with one of these two colours. A $Kempe$ $swap$ changes one colouring into another by interchanging the colours of the vertices in a Kempe chain. Two colourings are $Kempe$ $equivalent$ if each can be obtained from the other by a series of Kempe swaps; the set of Kempe equivalent colourings is called a $Kempe$ $class$. For a graph $G$, let $χ(G)$ denote its chromatic number and let $\mathcal{C}_{k}(G)$ denote the set of all $k$-colourings of $G$. We say $G$ is $Kempe$ $connected$ if for all $k\ge χ(G)$, $\mathcal{C}_{k}(G)$ forms a Kempe class. For a graph $H$, graph $G$ is called $H$-$free$ if no induced subgraph of $G$ is isomorphic to $H$. We prove that every $H$-free graph is Kempe connected if and only if $H$ is an induced subgraph of the path on four vertices, $P_4$. The graph 2$K_2$ consists of four vertices and two edges which are not adjacent. We prove that for all $p\ge 0$, there is a $k$-colourable 2$K_2$-free graph $G$ such that $\mathcal{C}_{k+p}(G)$ does not form a Kempe class.

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arXiv2024

Recoloring via modular decomposition

The reconfiguration graph of the $k$-colorings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_{k}(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell \geq χ(G)$+1. We demonstrate how to use the modular decomposition of a graph class to prove that the graphs in the class are recolorable. In particular, we prove that every ($P_5$, diamond)-free graph, every ($P_5$, house, bull)-free graph, and every ($P_5$, $C_5$, co-fork)-free graph is recolorable. A graph is prime if it cannot be decomposed by modular decomposition except into single vertices. For a prime graph $H$, we study the complexity of deciding if $H$ is $k$-colorable and the complexity of deciding if there exists a path between two given $k$-colorings in $R_{k}(H)$. Suppose $\mathcal{G}$ is a hereditary class of graphs. We prove that if every blowup of every prime graph in $\mathcal{G}$ is recolorable, then every graph in $\mathcal{G}$ is recolorable.

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arXiv2024

Frozen colourings in $2K_2$-free graphs

The \emph{reconfiguration graph of the $k$-colourings} of a graph $G$, denoted $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two vertices of $\mathcal{R}_k(G)$ are joined by an edge if the colourings of $G$ they correspond to differ in colour on exactly one vertex. A $k$-colouring of a graph $G$ is called \emph{frozen} if it is an isolated vertex in $\mathcal{R}_k(G)$; in other words, for every vertex $v \in V(G)$, $v$ is adjacent to a vertex of every colour different from its colour. A clique partition is a partition of the vertices of a graph into cliques. A clique partition is called a $k$-clique-partition if it contains at most $k$ cliques. Clearly, a $k$-colouring of a graph $G$ corresponds precisely to a $k$-clique-partition of its complement, $\overline{G}$. A $k$-clique-partition $\mathcal{Q}$ of a graph $H$ is called \emph{frozen} if for every vertex $v \in V(H)$, $v$ has a non-neighbour in each of the cliques of $\mathcal{Q}$ other than the one containing $v$. The cycle on four vertices, $C_4$, is sometimes called the \emph{square}; its complement is called $2K_2$. We give several infinite classes of $2K_2$-free graphs with frozen colourings. We give an operation which transforms a $k$-chromatic graph with a frozen $(k+1)$-colouring into a $(k+1)$-chromatic graph with a frozen $(k+2)$-colouring. Our operation preserves being $2K_2$-free. It follows that for all $k \ge 4$, there is a $k$-chromatic $2K_2$-free graph with a frozen $(k+1)$-colouring. We prove these results by studying frozen clique partitions in $C_4$-free graphs. We say a graph $G$ is \emph{recolourable} if $R_{\ell}(G)$ is connected for all $\ell$ greater than the chromatic number of $G$. We prove that every 3-chromatic $2K_2$-free graph is recolourable.

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arXiv2024

Recoloring some hereditary graph classes

The reconfiguration graph of the $k$-colorings, denoted $R_k(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_k(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell\geq χ(G)$+1. In this paper, we study the recolorability of several graph classes restricted by forbidden induced subgraphs. We prove some properties of a vertex-minimal graph $G$ which is not recolorable. We show that every (triangle, $H$)-free graph is recolorable if and only if every (paw, $H$)-free graph is recolorable. Every graph in the class of $(2K_2,\ H)$-free graphs, where $H$ is a 4-vertex graph except $P_4$ or $P_3$+$P_1$, is recolorable if $H$ is either a triangle, paw, claw, or diamond. Furthermore, we prove that every ($P_5$, $C_5$, house, co-banner)-free graph is recolorable.

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arXiv2023

Reconfiguration of vertex colouring and forbidden induced subgraphs

The reconfiguration graph of the $k$-colourings, denoted $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are adjacent in $\mathcal{R}_k(G)$ if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of $\mathcal{R}_{k+1}(G)$ for a $k$-colourable graph $G$ restricted by forbidden induced subgraphs. We show that $\mathcal{R}_{k+1}(G)$ is connected for every $k$-colourable $H$-free graph $G$ if and only if $H$ is an induced subgraph of $P_4$ or $P_3+P_1$. We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if $G$ is a $k$-colourable ($2K_2$, $C_4$)-free graph, then $\mathcal{R}_{k+1}(G)$ is connected with diameter at most $4n$. Furthermore, we show that $\mathcal{R}_{k+1}(G)$ is connected for every $k$-colourable ($P_5$, $C_4$)-free graph $G$.

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