Related papers: A Generalization of Gale's lemma

A graph $G$ is $k$-critical if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. We give a lower bound, $f_k(n) \geq…

Graph code is a linear code obtained from linear codes $C$ and a certain bipartite graph G. In this paper, I propose an expansion of the definition of graph code to general $l$-partite, and give its lower bound of minimum distance. I also…

We investigate possible list extensions of generalised majority edge colourings of graphs and provide several results concerning these. Given a graph $G=(V,E)$, a list assignment $L:E\to 2^C$ and some level of majority tolerance…

Given d \in (0,infty) let k_d be the smallest integer k such that d < 2k\log k. We prove that the chromatic number of a random graph G(n,d/n) is either k_d or k_d+1 almost surely.

\textit{Total Coloring} of a graph is a major coloring problem in combinatorial mathematics, introduced in the early $1960$s. A \textit{total coloring} of a graph $G$ is a map $f:V(G) \cup E(G) \rightarrow \mathcal{K}$, where $\mathcal{K}$…

The conflict-free chromatic index of a graph $G$ is the minimum number of colours in an edge colouring of $G$ such that the neighbourhood of every edge contains a colour appearing exactly once. Its vertex analogue is the conflict-free…

A new generalization of Fiedler's lemma is obtained by introducing the concept of the main function of a matrix. As applications, the universal spectra of the H-join, the spectra of the H-generalized join and the spectra of the generalized…

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo-Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by $m(G)+1$, where…

We obtain several new upper bounds of the odd graceful chromatic number of a graph $G$, which must be bipartite. Some of our bounds depend only on the number of the vertices of $G$ or the chromatic number of some graphs related to the…

The generalised colouring numbers $\mathrm{adm}_r(G)$, $\mathrm{col}_r(G)$, and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have…

We explore several generalizations of Whitney's theorem -- a classical formula for the chromatic polynomial of a graph. Following Stanley, we replace the chromatic polynomial by the chromatic symmetric function. Following Dohmen and Trinks,…

Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…

We study the linear list chromatic number, denoted $\lcl(G)$, of sparse graphs. The maximum average degree of a graph $G$, denoted $\mad(G)$, is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with…

Let $G$ be a graph with degree sequence $d_1\geq \ldots \geq d_n$. Slater proposed $s\ell(G)=\min\{ s: (d_1+1)+\cdots+(d_s+1)\geq n\}$ as a lower bound on the domination number $\gamma(G)$ of $G$. We show that deciding the equality of…

Haviv ({\em European Journal of Combinatorics}, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $\mathbb{R}$. We show that this holds…

The proper chromatic number $\Vec{\chi}(G)$ of a graph $G$ is the minimum $k$ such that there exists an orientation of the edges of $G$ with all vertex-outdegrees at most $k$ and such that for any adjacent vertices, the outdegrees are…

From the paper of the first author it follows that upper and lower bounds for $\gamma$-vector of a simple polytope imply the bounds for its $g$-,$h$- and $f$-vectors. In the paper of the second author it was obtained unimprovable upper and…

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…

A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees,…

We prove that for $n>k\geq 3$, if $G$ is an $n$-vertex graph with chromatic number $k$ but any its proper subgraph has smaller chromatic number, then $G$ contains at most $n-k+3$ copies of cliques of size $k-1$. This answers a problem of…