A graph G with maximum degree and edge chromatic number ′(G)> is edge-critical if ′(G− e)= for every edge e of G. New lower bounds are given for the average degree of an edge-critical graph, which improve on the best bounds previously known for most values of . Examples of edge-critical graphs are also given. In almost all cases, there remains a large gap between the best lower bound known and ...

Improper choosability of planar graphs has been widely studied. In particular, Škrekovski investigated the smallest integer gk such that every planar graph of girth at least gk is k-improper 2-choosable. He proved [9] that 6 ≤ g1 ≤ 9; 5 ≤ g2 ≤ 7; 5 ≤ g3 ≤ 6 and ∀k ≥ 4, gk = 5. In this paper, we study the greatest real M(k, l) such that every graph of maximum average degree less than M(k, l) is ...

It is proved that there are functions f (r) and N(r, s) such that for every positive integer r , s, each graph G with average degree d(G) = 2|E(G)|/|V (G)| ≥ f (r), and with at least N(r, s) vertices has a minor isomorphic to Kr,s or to the union of s disjoint copies of Kr . © 2005 Published by Elsevier Ltd

Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger’s Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have consider the average degree required to force an arbitrary graphH as a minor. Here, we strengthen (under certain conditions) a recent result by Reed...

Let G be a simple connected graph of order n with average 2degree sequence M1 ≥ M2 ≥ · · · ≥ Mn. Let ρ(G) denote the spectral radius of the adjacency matrix of G. We show that for each 1 ≤ l ≤ n and for any b ≥ max {di/dj | i ∼ j}, ρ(G) ≤ Ml − b+ √ (Ml + b)2 + 4b ∑l−1 i=1(Mi −Ml) 2 with equality if and only if M1 = M2 = · · · = Mn.

A graph H is said to be light in a family G of graphs if at least one member of G contains a copy of H and there exists an integer λ(H,G) such that each member G of G with a copy of H also has a copy K of H such that degG(v) ≤ λ(H,G) for all v ∈ V(K). In this paper, we study the light graphs in the class of graphs with small average degree, including the plane graphs with some restrictions on g...

‎Let $G$ be a graph without an isolated vertex‎, ‎the normalized Laplacian matrix $tilde{mathcal{L}}(G)$‎ ‎is defined as $tilde{mathcal{L}}(G)=mathcal{D}^{-frac{1}{2}}mathcal{L}(G)mathcal{D}^{-frac{1}{2}}$‎, where ‎$mathcal{D}$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎G‎$‎‎. ‎The eigenvalues of‎ $tilde{mathcal{L}}(G)$ are ‎called as ‎the ‎normalized Laplacian eigenva...

An incidence of an undirected graph G is a pair (v, e) where v is a vertex of G and e an edge of G incident with v. Two incidences (v, e) and (w, f) are adjacent if one of the following holds: (i) v = w, (ii) e = f or (iii) vw = e or f . An incidence coloring of G assigns a color to each incidence of G in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama et al. [...

We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in O(2d) time and exponential space for a constant εd depending only on d. Thus, we generalize the recent results of Björklund et al. [TALG 2012] on graphs of bounded degree. Then, we move to the problem of counting perfect matchings in a graph. We first present a simple algorit...

Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds onD(H) for almost all graphsH and proved that for ‘balanced’H random graphs provide extremal examples and determine the extremal function. Examples of ‘unbalanced graphs’ are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) a...