Related papers: Coloring some $(P_6,C_4)$-free graphs with $\Delta…

The Borodin-Kostochka Conjecture states that for a graph $G$, if $\Delta(G) \geq 9$ and $\omega(G) \leq \Delta(G)-1$, then $\chi(G)\leq\Delta(G) -1$. We prove the Borodin-Kostochka Conjecture for $(P_5, \text{gem})$-free graphs, i.e.,…

Borodin and Kostochka conjectured that every graph $G$ with maximum degree $\Delta \ge 9$ satisfies $\chi \le \max\{\omega, \Delta-1\}$. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main…

Borodin and Kostochka conjectured that every graph $G$ with $\Delta\ge9$ satisfies $\chi\le$ max $\{\omega, \Delta-1\}$. Gupta and Pradhan proved the Borodin-Kostochka conjecture for ($P_5$, $C_4$)-free graphs [{\em J. Appl. Math. Comp.}…

The Borodin-Kostochka Conjecture states that for a graph $G$, if $\Delta(G)\geq 9$, then $\chi(G)\leq\max\{\Delta(G)-1,\omega(G)\}$. In this paper, we prove the Borodin-Kostochka Conjecture holding for odd-hole-free graphs.

The Borodin--Kostochka conjecture states that every graph $G$ with maximum degree $\Delta(G)\ge 9$ satisfies $\chi(G)\le \max\{\omega(G),\Delta(G)-1\}$. In this paper, we verify this conjecture for graphs with sufficiently large maximum…

Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $\Delta(G)\ge 9$ and its clique number satisfies $\omega(G)\le \Delta(G)-1$, then its chromatic number satisfies $\chi(G) \le \Delta(G)-1$. We prove this…

Borodin & Kostochka conjectured that if maximum degree of a graph is greater than or equal to 9, then the chromatic number of the graph is less than or equal to maximum of {\omega} and maximum degree minus 1. Here we prove that this…

Problem of finding an optimal upper bound for the chromatic no. of a graph is still open and very hard. Borodin and Kostochka Conjecture is still open and if proved will improve Brook bound on Chromatic no. of a graph. Here we prove Borodin…

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \left\{\omega,…

Problem of finding an optimal upper bound for the chromatic no. of even 3K1-free graphs is still open and pretty hard. Here we prove Borodin & Kostochka Conjecture for 4K1-free graphs G i.e. If maximum degree of a {4 Times K1}-free graph is…

An equivalent version of the Borodin-Kostochka Conjecture, due to Cranston and Rabern, says that any graph with $\chi = \Delta = 9$ contains $K_3 \lor E_6$ as a subgraph. Here we prove several results in support of this conjecture, where…

Let $K_p$ be a complete graph of order $p\geq 2$. A $K_p$-free $k$-coloring of a graph $H$ is a partition of $V(H)$ into $V_1, V_2\ldots,V_k$ such that $H[V_i]$ does not contain $K_p$ for each $i\leq k $. In 1977 Borodin and Kostochka…

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon, Krivelevich,…

Let $\chi'_\subset(G)$ be the least number of colours necessary to properly colour the edges of a graph $G$ with minimum degree $\delta\geq 2$ so that the set of colours incident with any vertex is not contained in a set of colours incident…

For a given graph $H$ and the graphical properties $P_1, P_2,\ldots,P_k$, a graph $H$ is said to be $(V_1, V_2,\ldots,V_k)$-partitionable if there exists a partition of $V(H)$ into $k$-sets $V_1, V_2\ldots,V_k$, such that for each…

The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer $h$, the proper $h$-conflict-free chromatic number of a graph $G$, denoted $\chi_{pcf}^h(G)$, is the minimum $k$ such that $G$…

A $(c_1,c_2,...,c_k)$-coloring of $G$ is a mapping $\varphi:V(G)\mapsto\{1,2,...,k\}$ such that for every $i,1 \leq i \leq k$, $G[V_i]$ has maximum degree at most $c_i$, where $G[V_i]$ denotes the subgraph induced by the vertices colored…

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ whenever $G$ is an $F$-free graph of maximum degree $\Delta$. The…

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 cyclic coloring of a plane graph $G$ is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph $G$ is its cyclic chromatic number…