相关论文: Structure and coloring of some ($P_7,C_4$)-free gr…

Let $P_t$ and $C_t$ be a path and a cycle on $t$ vertices, respectively. In 2021, Choudum {\em et al.} [Disc. Math. 344 (2021) 112244] determined the structures of $(P_7,C_7,C_4$, diamond)-free and $(P_7,C_7,C_4$, gem)-frees, and gave…

We denote a path on $t$ vertices as $P_t$ and a cycle on $t$ vertices as $C_t$. For two vertex-disjoint graphs $G_1$ and $G_2$, the {\em union} $G_1\cup G_2$ is the graph with $V(G_1\cup G_2)=V(G_1)\cup V(G_2)$ and $E(G_1\cup…

For a graph $G$, let $\chi(G)$ and $\omega(G)$ 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…

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 $\chi(G) = 6$. This generalizes some known results in the literature.

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_s$ be the path on $t$ vertices and the cycle on $s$ vertices, respectively. In this paper we show…

A diamond is the graph that is obtained from removing an edge from the complete graph on $4$ vertices. A ($C_4$,diamond)-free graph is a graph that does not contain a diamond or a cycle on four vertices as induced subgraphs. Let $G$ be a…

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices and $K_t$ be the complete graph on $t$ vertices. The diamond is the…

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…

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…

For a graph $G$, $\chi(G)$ $(\omega(G))$ 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…

In this paper, we establish an optimal $\chi$-binding function for $(P_2\cup P_4,\text{ diamond})$-free graphs. We prove that for any graph $G$ in this class, $\chi(G)\le 4$ when $\omega(G)=2$, $\chi(G)\le 6$ when $\omega(G)=3$, and…

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…

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is ($P_6$, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the…

This article foucuses on $(P_3\cup P_2,K_4)$-free graph. In this paper, we prove that if G is $(P_3\cup P_2,K_4)$-free, then $\chi(G)\le 7$. We then use our result to obtain the upper bound of order and chromatic number of…

As usual, $P_n$ ($n \geq 1$) denotes the path on $n$ vertices, and $C_n$ ($n \geq 3$) denotes the cycle on $n$ vertices. For a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is…

We consider the class of (C4, diamond)-free graphs; graphs in this class do not contain a C4 or a diamond as an induced subgraph. We provide an efficient recognition algorithm for this class. We count the number of maximal cliques in a (C4,…

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…

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…

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in…

A class ${\cal G}$ of graphs is $\chi$-{\em polydet} if ${\cal G}$ has a polynomial binding function $f$ and there is a polynomial time algorithm to determine an $f(\omega(G))$-coloring of $G\in {\cal G}$. Let $P_t$ and $C_t$ denote a path…