Weak degeneracy of graphs

Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every $d$-degenerate graph is also weakly $d$-degenerate. On the other hand, if $G$ is weakly $d$-degenerate, then $\chi(G) \leq d + 1$ (and, moreover, the same bound holds for the list-chromatic and even the DP-chromatic number of $G$). It turns out that several upper bounds in graph coloring theory can be phrased in terms of weak degeneracy. For example, we show that planar graphs are weakly $4$-degenerate, which implies Thomassen's famous theorem that planar graphs are $5$-list-colorable. We also prove a version of Brooks's theorem for weak degeneracy: a connected graph $G$ of maximum degree $d \geq 3$ is weakly $(d-1)$-degenerate unless $G \cong K_{d + 1}$. (By contrast, all $d$-regular graphs have degeneracy $d$.) We actually prove an even stronger result, namely that for every $d \geq 3$, there is $\epsilon > 0$ such that if $G$ is a graph of weak degeneracy at least $d$, then either $G$ contains a $(d+1)$-clique or the maximum average degree of $G$ is at least $d + \epsilon$. Finally, we show that graphs of maximum degree $d$ and either of girth at least $5$ or of bounded chromatic number are weakly $(d - \Omega(\sqrt{d}))$-degenerate, which is best possible up to the value of the implied constant.Comment: 21 p

Graph colouring for office blocks

The increasing prevalence of WLAN (wireless networks) introduces the potential of electronic information leakage from one company's territory in an office block, to others due to the long-ranged nature of such communications. BAE Systems have developed a system ('stealthy wallpaper') which can block a single frequency range from being transmitted through a treated wall or ceiling to the neighbour. The problem posed to the Study Group was to investigate the maximum number of frequencies ensure the building is secure. The Study group found that this upper bound does not exist, so they were asked to find what are "good design-rules" so that an upper limit exists

Strong chromatic index of sparse graphs

A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $\chi_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In this note we prove that $\chi_{s}^{\prime}(G)\leq (4k-1)\Delta (G)-k(2k+1)+1$ for every $k$-degenerate graph $G$. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that $$ for any chordless graph $G$. Both bounds remain valid for the list version of the strong edge coloring of these graphs