A graph \( G \) with maximum degree \( \Delta \) and edge chromatic number \( \chi'(G) > \Delta \) is \emph{edge-\(\Delta\)-critical} if \( \chi'(G-e) = \Delta \) for each \( e \in E(G) \). In this article, we provide a new proof of adjacency Lemmas on edge-critical graphs such that Vizing’s adjacency lemma becomes a corollary of our results.

This paper surveys recent results for flag enumeration of polytopes, Bruhat graphs, balanced digraphs, Whitney stratified spaces and quasi-graded posets.

A bipancyclic graph on \( v \) vertices is a bipartite graph that contains, as subgraphs, cycles of length \( n \) for every even integer \( n \) such that \( 4 \leq n \leq v \). Such a graph is uniquely bipancyclic if it contains exactly one subgraph of each permissible length.

In this paper, we find all uniquely bipancyclic graphs on 30 or fewer vertices.

A balanced complete bipartite graph is a complete bipartite graph where the degrees of its vertices differ by at most 1. In a red-blue-green coloring of the edges of a graph \( G \), every edge of \( G \) is colored red, blue, or green. For three graphs \( F_1 \), \( F_2 \), and \( F_3 \), the 2-Ramsey number \( R_2(F_1, F_2, F_3) \) of \( F_1 \), \( F_2 \), and \( F_3 \), if it exists, is the smallest order of a balanced complete bipartite graph \( G \) such that every red-blue-green coloring of the edges of \( G \) contains a red \( F_1 \), a blue \( F_2 \), or a green \( F_3 \). In this note, we determine that

\[
20 \leq R_2(C_4, C_4, C_4) \leq 21.
\]

A Hamiltonian graph \( G \) is said to be \(\ell\)-path-Hamiltonian, where \(\ell\) is a positive integer less than or equal to the order of \( G \), if every path of order \(\ell\) in \( G \) is a subpath of some Hamiltonian cycle in \( G \). The Hamiltonian cycle extension number of \( G \) is the maximum positive integer \(\ell\) for which every path of order \(\ell\) or less is a subpath of some Hamiltonian cycle in \( G \). If the order of \( G \) equals \( n \), then it is known that \( \text{hce}(G) = n \) if and only if \( G \) is a cycle or a regular complete bipartite graph (when \( n \) is even) or a complete graph. We present a complete characterization of Hamiltonian graphs of order \( n \) that are \(\ell\)-path-Hamiltonian for each \(\ell \in \{n-3, n-2, n-1, n\}\).

Let \( G \) be an edge-colored connected graph. A path \( P \) is a proper path in \( G \) if no two adjacent edges of \( P \) are colored the same. An edge coloring is a proper-path coloring of \( G \) if every pair \( u, v \) of distinct vertices of \( G \) is connected by a proper \( u-v \) path in \( G \). The minimum number of colors required for a proper-path coloring of \( G \) is the proper connection number \( \text{pc}(G) \) of \( G \). We study proper-path colorings in those graphs obtained by some well-known graph operations, namely line graphs, powers of graphs, coronas of graphs, and vertex or edge deletions. Proper connection numbers are determined for all iterated line graphs and powers of a given connected graph. For a connected graph \( G \), sharp lower and upper bounds are established for the proper connection number of (i) the \( k \)-iterated corona of \( G \) in terms of \( \text{pc}(G) \) and \( k \), and (ii) the vertex or edge deletion graphs \( G-v \) and \( G-e \), where \( v \) is a non-cut-vertex of \( G \) and \( e \) is a non-bridge of \( G \), in terms of \( \text{pc}(G) \) and the degree of \( v \). Other results and open questions are also presented.

A red-blue coloring of a graph \( G \) is an edge coloring of \( G \) in which every edge of \( G \) is colored red or blue. Let \( F \) be a connected graph of size 2 or more with a red-blue coloring, at least one edge of each color, where some blue edge of \( F \) is designated as the root of \( F \). Such an edge-colored graph \( F \) is called a color frame. An \( F \)-coloring of a graph \( G \) is a red-blue coloring of \( G \) in which every blue edge of \( G \) is the root edge of a copy of \( F \) in \( G \). The \( F \)-chromatic index \( \chi_F'(G) \) of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). A minimal \( F \)-coloring of \( G \) is an \( F \)-coloring with the property that if any red edge of \( G \) is re-colored blue, then the resulting red-blue coloring of \( G \) is not an \( F \)-coloring of \( G \). The maximum number of red edges in a minimal \( F \)-coloring of \( G \) is the upper \( F \)-chromatic index \( \chi_F”(G) \) of \( G \). For integers \( k \) and \( m \) with \( 1 \leq k < m \) and \( m \geq 3 \), let \( S_{k,m} \) be the color frame of the star \( K_{1,m} \) of size \( m \) such that \( S_{k,m} \) has exactly \( k \) red edges and \( m-k \) blue edges. For a positive integer \( k \), a set \( X \) of edges of a graph \( G \) is a \( \Delta_k \)-set if \( \Delta(G[X]) = k \), where \( G[X] \) is the subgraph of \( G \) induced by \( X \). The maximum size of a \( \Delta_k \)-set in \( G \) is referred to as the \( k \)-matching number of \( G \) and is denoted by \( a_k'(G) \). A \( \Delta_k \)-set \( X \) is maximal if \( X \cup \{e\} \) is not a \( \Delta_k \)-set for every \( e \in E(G) – X \). The minimum size of a maximal \( \Delta_k \)-set of \( G \) is the lower \( k \)-matching number of \( G \) and is denoted by \( a_k''(G) \). In this paper, we consider \( S_{k,m} \)-colorings of a graph and study relations between \( S_{k,m} \)-colorings and \( \Delta_k \)-sets in graphs. Bounds are established for the \( S_{k,m} \)-chromatic indexes \( \chi_{S_{k,m}}'(G) \) and \( \chi_{S_{k,m}}''(G) \) of a graph \( G \) in terms of the \( k \)-matching numbers \( a_k'(G) \) and \( a_k''(G) \) of the graph. Other results and questions are also presented.

Let \( G \) be a Hamiltonian graph of order \( n \geq 3 \). For an integer \(\ell\) with \(1 \leq \ell \leq n\), the graph \( G \) is \(\ell\)-path-Hamiltonian if every path of order \(\ell\) lies on a Hamiltonian cycle in \( G \). The Hamiltonian cycle extension number of \( G \) is the maximum positive integer \(\ell\) for which every path of order \(\ell\) or less lies on a Hamiltonian cycle of \( G \). For an integer \(\ell\) with \(2 \leq \ell \leq n-1\), the graph \( G \) is \(\ell\)-path-pancyclic if every path of order \(\ell\) in \( G \) lies on a cycle of every length from \(\ell+1\) to \(n\). (Thus, a \(2\)-path-pancyclic graph is edge-pancyclic.) A graph \( G \) of order \( n \geq 3 \) is path-pancyclic if \( G \) is \(\ell\)-path-pancyclic for each integer \(\ell\) with \(2 \leq \ell \leq n-1\). In this paper, we present a brief survey of known results on these two parameters and investigate the \(\ell\)-path-Hamiltonian graphs and \(\ell\)-path-pancyclic graphs having small minimum degree and small values of \(\ell\). Furthermore, highly path-pancyclic graphs are characterized and several well-known classes of \(2\)-path-pancyclic graphs are determined. The relationship among these two parameters and other well-known Hamiltonian parameters is investigated along with some open questions in this area of research.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). A \((p, q)\)-graph \( G = (V, E) \) is said to be AL(\(k\))-traversal if there exists a sequence of vertices \((v_1, v_2, \ldots, v_p)\) such that for each \( i = 1, 2, \ldots, p-1 \), the distance between \( v_i \) and \( v_{i+1} \) is equal to \( k \). We call a graph \( G \) a 2-steps Hamiltonian graph if it has an AL(2)-traversal in \( G \) and \( d(v_p, v_1) = 2 \). In this paper, we characterize some cubic graphs that are 2-steps Hamiltonian. We show that no forbidden subgraph characterization for non-2-steps-Hamiltonian cubic graphs is available by demonstrating that every cubic graph is a homeomorphic subgraph of a non-2-steps Hamiltonian cubic graph.

For a graph \(G = (V, E)\) and a coloring \(f : V(G) \to \mathbb{Z}_2\), let \(v_f(i) = |f^{-1}(i)|\). \(f\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). The coloring \(f : V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f_+ : E(G) \to \mathbb{Z}_2\) defined by \(f_+(xy) = |f(x) – f(y)|\), for all \(xy \in E(G)\). Let \(e_f(i) = |f_+^{-1}(i)|\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined by
\[
FI(G) = \{ |e_f(1) – e_f(0)| : f \text{ is a friendly vertex labeling of } G \}.
\]

In this paper, we determine the friendly index set of certain classes of trees and introduce a few classes of fully cordial trees.