arXiv subjects

T. Karthick

Publications and source records attributed to T. Karthick.

At least 19 recordsLinked to original sources

arXiv2026

Vertex-critical co-gem-free graphs

Given a graph $G$, let $\chi(G)$ denote the chromatic number of $G$. For $k\in \mathbb{N}$, a graph $G$ is $k$-$vertex$-$critical$ if $\chi(G)=k$ and $\chi(G-v)< k$ 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$\}$.

math.CO↗

arXiv2025

Reconfiguration graph for vertex colorings for ($P_2$+$P_3$, $C_4$)-free graphs

For a graph $G$, let $\chi(G)$ denote the chromatic number of $G$. Given a graph $G$, the $reconfiguration$ $graph$ $for$ $the$ $k$-$colorings$ of $G$, denoted by ${\cal R}_k(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two $k$-colorings are joined by an edge if they differ on exactly one vertex of $G$. A graph $G$ is $k$-$mixing$ if ${\cal R}_k(G)$ is connected, and is $recolorable$ if it is $k$-mixing for all $k> \chi(G)$. In this paper, we give a complete characterization of $(P_2+P_3, C_4)$-free graphs that are recolorable. Moreover, we show that if $G$ is a recolorable $(P_2+P_3, C_4)$-free graph, then for any $k >\chi(G)$, the diameter of ${\cal R}_k(G)$ is at most 2$n^{2}$. Furthermore, we show that if $G$ is a ($P_2+P_3, C_4$)-free graph on $n$ vertices with degeneracy $\rho(G)$, then for all $k > \rho(G)+ 1$, the diameter of ${\cal R}_k(G)$ is at most $O(n^2)$. This confirms a conjecture of Cereceda for the class of ($P_2+P_3, C_4$)-free graphs. These results generalize some known results available in the literature.

math.CO↗

arXiv2025

On near optimal colorable graphs

A class of graphs $\cal G$ is said to be \emph{near optimal colorable} if there exists a constant $c\in \mathbb{N}$ such that every graph $G\in \cal G$ satisfies $\chi(G) \leq \max\{c, \omega(G)\}$, where $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. The class of near optimal colorable graphs is an important subclass of the class of $\chi$-bounded graphs which is well-studied in the literature. In this paper, we show that the class of ($F, K_4-e$)-free graphs is near optimal colorable, where $F\in \{P_1+2P_2,2P_1+P_3,3P_1+P_2\}$ and the graph $K_4-e$ is commonly referred as the {\em diamond}. This partially answers a question of Ju and Huang [Theoretical Computer Science 993 (2024) Article No.: 114465] and is related to a question of Schiermeyer (unpublished). Furthermore, using these results with some earlier known results, we also provide an alternate proof to the fact that the \textsc{Chromatic Number} problem for the class of ($F, K_4-e$)-free graphs is solvable in polynomial time, where $F\in \{P_1+2P_2,2P_1+P_3,3P_1+P_2\}$.

cs.DM↗

arXiv2025

($P_2+P_4$, $K_4-e$)-free graphs are nearly $\omega$-colorable

For a graph $G$, $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. In this paper, we show the following results: (i) If $G$ is a ($P_2+P_4$, $K_4-e$)-free graph with $\omega(G)\geq 3$, then $\chi(G)\leq \max\{6, \omega(G)\}$, and the bound is tight for each $\omega(G)\notin \{4,5\}$. (ii) If $G$ is a ($P_2+P_4$, $K_4-e$)-free graph with $\omega(G)= 4$, then $\chi(G)= 4$. These results extend the chromatic bounds known for the class of ($P_2+P_2$, $K_4-e$)-free graphs and for the class of ($P_2+P_3$, $K_4-e$)-free graphs, improve the bound of Chen and Zhang [arXiv:2412.14524 [math.CO], 2024] given for the class of ($P_2+P_4$, $K_4-e$)-free graphs, partially answer a question of Ju and the third author [Theor. Comp. Sci. 993 (2024) Article No.: 114465] on `near optimal colorable graphs', and a question of Schiermeyer (unpublished) on the chromatic bound for ($P_7$, $K_4-e$)-free graphs.

math.CO↗

arXiv2024

An optimal chromatic bound for ($P_2+P_3$, gem)-free graphs

Given a graph $G$, the parameters $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and the clique number of $G$. A function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(1) = 1$ and $f(x) \geq x$, for all $x \in \mathbb{N}$ is called a $\chi$-binding function for the given class of graphs $\cal{G}$ if every $G \in \cal{G}$ satisfies $\chi(G) \leq f(\omega(G))$, and the \emph{smallest $\chi$-binding function} $f^*$ for $\cal{G}$ is defined as $f^*(x) := \max\{\chi(G)\mid G\in {\cal G} \mbox{ and } \omega(G)=x\}$. In general, the problem of obtaining the smallest $\chi$-binding function for the given class of graphs seems to be extremely hard, and only a few classes of graphs are studied in this direction. In this paper, we study the class of ($P_2+ P_3$, gem)-free graphs, and prove that the function $\phi:\mathbb{N}\rightarrow \mathbb{N}$ defined by $\phi(1)=1$, $\phi(2)=4$, $\phi(3)=6$ and $\phi(x)=\left\lceil\frac{1}{4}(5x-1)\right\rceil$, for $x\geq 4$ is the smallest $\chi$-binding function for the class of ($P_2+ P_3$, gem)-free graphs.

math.CO↗

arXiv2023

On graphs with no induced $P_5$ or $K_5-e$

In this paper, we are interested in some problems related to chromatic number and clique number for the class of $(P_5,K_5-e)$-free graphs, and prove the following. $(a)$ If $G$ is a connected ($P_5,K_5-e$)-free graph with $\omega(G)\geq 7$, then either $G$ is the complement of a bipartite graph or $G$ has a clique cut-set. Moreover, there is a connected ($P_5,K_5-e$)-free imperfect graph $H$ with $\omega(H)=6$ and has no clique cut-set. This strengthens a result of Malyshev and Lobanova [Disc. Appl. Math. 219 (2017) 158--166]. $(b)$ If $G$ is a ($P_5,K_5-e$)-free graph with $\omega(G)\geq 4$, then $\chi(G)\leq \max\{7, \omega(G)\}$. Moreover, the bound is tight when $\omega(G)\notin \{4,5,6\}$. This result together with known results partially answers a question of Ju and Huang [arXiv:2303.18003 [math.CO] 2023], and also improves a result of Xu [Manuscript 2022]. While the "Chromatic Number Problem" is known to be $NP$-hard for the class of $P_5$-free graphs, our results together with some known results imply that the "Chromatic Number Problem" can be solved in polynomial time for the class of ($P_5,K_5-e$)-free graphs which may be independent interest.

math.CO↗

arXiv2022

Optimal chromatic bound for ($P_2+P_3$, $\bar{P_2+ P_3}$)-free graphs

For a graph $G$, let $χ(G)$ ($ω(G)$) denote its chromatic (clique) number. A $P_2+P_3$ is the graph obtained by taking the disjoint union of a two-vertex path $P_2$ and a three-vertex path $P_3$. A $\bar{P_2+P_3}$ is the complement graph of a $P_2+P_3$. In this paper, we study the class of ($P_2+P_3$, $\bar{P_2+P_3}$)-free graphs and show that every such graph $G$ with $ω(G)\geq 3$ satisfies $χ(G)\leq \max \{ω(G)+3, \lfloor\frac{3}{2} ω(G) \rfloor-1 \}$. Moreover, the bound is tight. Indeed, for any $k\in {\mathbb N}$ and $k\geq 3$, there is a ($P_2+P_3$, $\bar{P_2+P_3}$)-free graph $G$ such that $ω(G)=k$ and $χ(G)=\max\{k+3, \lfloor\frac{3}{2} k \rfloor-1 \}$.

math.CO↗

arXiv2022

Coloring ($P_5$, kite)-free graphs

Let $P_n$ and $K_n$ denote the induced path and complete graph on $n$ vertices, respectively. The {\em kite} is the graph obtained from a $P_4$ by adding a vertex and making it adjacent to all vertices in the $P_4$ except one vertex with degree 1. A graph is ($P_5$, kite)-free if it has no induced subgraph isomorphic to a $P_5$ or a kite. For a graph $G$, the chromatic number of $G$ (denoted by $χ(G)$) is the minimum number of colors needed to color the vertices of $G$ such that no two adjacent vertices receive the same color, and the clique number of $G$ is the size of a largest clique in $G$. Here, we are interested in the class of ($P_5$, kite)-free graphs with small clique number. It is known that every ($P_5$,~kite, $K_3$)-free graph $G$ satisfies $χ(G)\leq 3$, every ($P_5$,~kite, $K_4$)-free graph $G$ satisfies $χ(G)\leq 4$, and that every ($P_5$,~kite, $K_5$)-free graph $G$ satisfies $χ(G)\leq 6$. In this paper, we showed the following: $\bullet$ Every ($P_5$, kite, $K_6$)-free graph $G$ satisfies $χ(G)\leq 7$. $\bullet$ Every ($P_5$, kite, $K_7$)-free graph $G$ satisfies $χ(G)\leq 9$. We also give examples to show that the above bounds are tight.

math.CO↗

arXiv2021

Coloring of ($P_5$, $4$-wheel)-free graphs

For a graph $G$, $χ(G)$ $(ω(G))$ denote its chromatic (clique) number. A $P_5$ is the chordless path on five vertices, and a $4$-$wheel$ is the graph consisting of a chordless cycle on four vertices $C_4$ plus an additional vertex adjacent to all the vertices of the $C_4$. In this paper, we show that every ($P_5$, $4$-wheel)-free graph $G$ satisfies $χ(G)\leq \frac{3}{2}ω(G)$. Moreover, this bound is almost tight. That is, there is a class of ($P_5$, $4$-wheel)-free graphs $\cal L$ such that every graph $H\in \cal L$ satisfies $χ(H)\geq\frac{10}{7}ω(H)$. This generalizes/improves several previously known results in the literature.

math.CO↗

arXiv2021

Coloring graph classes with no induced fork via perfect divisibility

For a graph $G$, $χ(G)$ will denote its chromatic number, and $ω(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. An integer-valued function $f$ is called a $χ$-binding function for a hereditary class of graphs $\cal C$ if $χ(G) \leq f(ω(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a polynomial $χ$-binding function for the class of fork-free graphs is open. In this paper, we study the structure of some classes of fork-free graphs; in particular, we study the class of (fork,$F$)-free graphs $\cal G$ in the context of perfect divisibility, where $F$ is a graph on five vertices with a stable set of size three, and show that every $G\in \cal G$ satisfies $χ(G)\leq ω(G)^2$. We also note that the class $\cal G$ does not admit a linear $χ$-binding function.

math.CO↗

arXiv2019

Structural domination and coloring of some ($P_7, C_7$)-free graphs

We show that every connected induced subgraph of a graph $G$ is dominated by an induced connected split graph if and only if $G$ is $\cal{C}$-free, where $\cal{C}$ is a set of six graphs which includes $P_7$ and $C_7$, and each containing an induced $P_5$. A similar characterisation is shown for the class of graphs which are dominated by induced complete split graphs. Motivated by these results, we study structural descriptions of some classes of $\cal{C}$-free graphs. In particular, we give structural descriptions for the class of ($P_7$,$C_7$,$C_4$,gem)-free graphs and for the class of ($P_7$,$C_7$,$C_4$,diamond)-free graphs. Using these results, we show that every ($P_7$,$C_7$,$C_4$,gem)-free graph $G$ satisfies $χ(G) \leq 2ω(G)-1$, and that every ($P_7$,$C_7$,$C_4$,diamond)-free graph $H$ satisfies $χ(H) \leq ω(H)+1$. These two upper bounds are tight for any subgraph of the Petersen graph containing a $C_5$.

math.CO↗

arXiv2019

On graphs with no induced five-vertex path or paraglider

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,\,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. For a positive integer $t$, $P_t$ is the chordless path on $t$ vertices. A paraglider is the graph that consists of a chorless cycle $C_4$ plus a vertex adjacent to three vertices of the $C_4$. In this paper, we study the structure of ($P_5$, paraglider)-free graphs, and show that every such graph $G$ satisfies $χ(G)\le \lceil \frac{3}{2}ω(G) \rceil$, where $χ(G)$ and $ω(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the Clebsch graph on 16 vertices. More strongly, we completely characterize all the ($P_5$, paraglider)-free graphs $G$ that satisfies $χ(G)> \frac{3}{2}ω(G)$. We also construct an infinite family of ($P_5$, paraglider)-free graphs such that every graph $G$ in the family has $χ(G)=\lceil \frac{3}{2}ω(G) \rceil-1$. This shows that our upper bound is optimal up to an additive constant and that there is no $(\frac{3}{2}-ε)$-approximation algorithm to the chromatic number of ($P_5$, paraglider)-free graphs for any $ε>0$.

math.CO↗

arXiv2019

Square-free graphs with no six-vertex induced path

We elucidate the structure of $(P_6,C_4)$-free graphs by showing that every such graph either has a clique cutset, or a universal vertex, or belongs to several special classes of graphs. Using this result, we show that for any $(P_6,C_4)$-free graph $G$, $\lceil\frac{5ω(G)}{4}\rceil$ and $\lceil\frac{Δ(G) + ω(G) +1}{2}\rceil$ are tight upper bounds for the chromatic number of $G$. Moreover, our structural results imply that every ($P_6$,$C_4$)-free graph with no clique cutset has bounded clique-width, and thus the existence of a polynomial-time algorithm that computes the chromatic number (or stability number) of any $(P_6,C_4)$-free graph.

cs.DM↗

arXiv2019

Coloring graphs with no induced five-vertex path or gem

For a graph $G$, let $χ(G)$ and $ω(G)$ respectively denote the chromatic number and clique number of $G$. We give an explicit structural description of ($P_5$,gem)-free graphs, and show that every such graph $G$ satisfies $χ(G)\le \lceil\frac{5ω(G)}{4}\rceil$. Moreover, this bound is best possible.

math.CO↗

arXiv2018

Polynomial Cases for the Vertex Coloring Problem

The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for ($P_5$, dart)-free graphs, ($P_5$, banner)-free graphs, ($P_5$, bull)-free graphs, and (fork, bull)-free graphs.

math.CO↗

arXiv2018

Chromatic bounds for some classes of $2K_2$-free graphs

A hereditary class $\mathcal{G}$ of graphs is $χ$-bounded if there is a $χ$-binding function, say $f$ such that $χ(G) \leq f(ω(G))$, for every $G \in \cal{G}$, where $χ(G)$ ($ω(G)$) denote the chromatic (clique) number of $G$. It is known that for every $2K_2$-free graph $G$, $χ(G) \leq \binom{ω(G)+1}{2}$, and the class of ($2K_2, 3K_1$)-free graphs does not admit a linear $χ$-binding function. In this paper, we are interested in classes of $2K_2$-free graphs that admit a linear $χ$-binding function. We show that the class of ($2K_2, H$)-free graphs, where $H\in \{K_1+P_4, K_1+C_4, \overline{P_2\cup P_3}, HVN, K_5-e, K_5\}$ admits a linear $χ$-binding function. Also, we show that some superclasses of $2K_2$-free graphs are $χ$-bounded.

cs.DM↗

arXiv2017

Coloring ($P_6$, diamond, $K_4$)-free graphs

We show that every ($P_6$, diamond, $K_4$)-free graph is $6$-colorable. Moreover, we give an example of a ($P_6$, diamond, $K_4$)-free graph $G$ with $χ(G) = 6$. This generalizes some known results in the literature.

math.CO↗

arXiv2016

Independent Sets in Classes Related to Chair/Fork-free Graphs

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. MWIS is known to be $NP$-complete in general, even under various restrictions. Let $S_{i,j,k}$ be the graph consisting of three induced paths of lengths $i, j, k$ with a common initial vertex. The complexity of the MWIS problem for $S_{1, 2, 2}$-free graphs, and for $S_{1, 1, 3}$-free graphs are open. In this paper, we show that the MWIS problem can solved in polynomial time for ($S_{1, 2, 2}$, $S_{1, 1, 3}$, co-chair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This extends some known results in the literature.

cs.DM↗